Step 1: Initial Setup & Theoretical Foundation

Let the given mathematical expression be defined as a polynomial function $P(x)$ in terms of the single variable $x$:

$P(x) = 5x^3 + 4x^2 + 7x$

[Per the Fundamental Theorem of Algebra and standard algebraic definitions, the degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the expression, provided that the coefficient of that term is strictly non-zero.]

Step 2: Term-by-Term Exponent Analysis

To determine the degree, we must decompose the polynomial into its constituent terms and isolate the exponent of the variable $x$ in each term.

• First Term: $5x^3$
  The variable is $x$, and its exponent is $3$. The coefficient is $5$ (where $5 \neq 0$).
• Second Term: $4x^2$
  The variable is $x$, and its exponent is $2$. The coefficient is $4$ (where $4 \neq 0$).
• Third Term: $7x$
  By the laws of exponents, any variable without an explicitly written power has an implicit exponent of $1$. Thus, $7x$ can be rewritten as $7x^1$. The exponent is $1$. The coefficient is $7$ (where $7 \neq 0$).

Step 3: Visualizing the Polynomial Structure

The following diagram isolates each term, highlighting the exponents to visually confirm the maximum power.

Step 4: Determination of the Maximum Exponent

We extract the set of all exponents present in the polynomial $P(x)$:

$E = \{3, 2, 1\}$

Comparing these values, we find the maximum element in the set:

$\max(3, 2, 1) = 3$

Because the coefficient associated with $x^3$ is $5$ (which satisfies the condition $5 \neq 0$), the highest power is validated as the degree of the polynomial.

Final Solution: The degree of the polynomial $5x^3 + 4x^2 + 7x$ is $3$.

Theoretical Foundation & Definitions

To construct the required algebraic expressions, we must first establish the rigorous definitions of the polynomial classifications based on their terms and degrees [Per the Fundamental Definitions of Polynomial Algebra]:

• Polynomial: An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
• Term: A single mathematical expression separated by addition ($+$) or subtraction ($-$) signs.
• Monomial: A polynomial consisting of exactly one non-zero term.
• Binomial: A polynomial consisting of exactly two non-zero terms.
• Degree of a Polynomial in One Variable: The highest exponent (power) of the variable present in the polynomial with a non-zero coefficient.

Step 1: Constructing a Binomial of Degree 35

Based on the definitions, a binomial of degree $35$ must satisfy two independent conditions:

1. It must contain exactly two terms.
2. The highest power of the variable must be exactly $35$.

The general form of such a binomial in variable $x$ can be written as:

$P(x) = ax^{35} + bx^k$

Where:

• $a$ and $b$ are non-zero real numbers ($a \neq 0, b \neq 0$).
• $k$ is a non-negative integer strictly less than $35$ ($0 \le k < 35$).

By selecting $a = 3$, $b = -4$, and $k = 2$, we generate a valid example:

$3x^{35} - 4x^2$

[Justification: The expression has two terms ($3x^{35}$ and $-4x^2$), making it a binomial. The highest exponent on the variable $x$ is $35$, making its degree $35$.]

Step 2: Constructing a Monomial of Degree 100

A monomial of degree $100$ must satisfy the following conditions:

1. It must contain exactly one term.
2. The power of the variable in that single term must be exactly $100$.

The general form of such a monomial in variable $y$ can be written as:

$Q(y) = cy^{100}$

Where:

• $c$ is any non-zero real number ($c \neq 0$).

By selecting $c = \sqrt{5}$, we generate a valid example:

$\sqrt{5}y^{100}$

[Justification: The expression consists of a single term ($\sqrt{5}y^{100}$), making it a monomial. The exponent on the variable $y$ is $100$, making its degree $100$.]

Visual Analysis of Polynomial Anatomy

The following diagram illustrates the structural components of the constructed polynomials, verifying their terms and degrees.

Final Solution:
An example of a binomial of degree 35 is $3x^{35} - 4x^2$.
An example of a monomial of degree 100 is $\sqrt{5}y^{100}$.
(Note: Infinite valid examples exist by altering the non-zero coefficients or the lower-degree terms in the binomial).

Initial Setup & Algebraic Definition

Given the algebraic expression: $P(x) = 4x^2 - 3x + 7$

[Per the Fundamental Theorem of Algebra and Polynomial Definitions], an algebraic expression is classified as a polynomial in one variable if and only if it satisfies two strict mathematical conditions:

• Condition 1 (Single Variable): The expression must contain only one distinct variable symbol (e.g., only $x$, only $y$, or only $z$).
• Condition 2 (Non-Negative Integer Exponents): The exponent of the variable in every single term must be a whole number ($\mathbb{W} = \{0, 1, 2, 3, \dots\}$). Fractional, negative, or irrational exponents disqualify the expression from being a polynomial.

Step 1: Evaluating the Variable Count

By inspecting the expression $4x^2 - 3x + 7$, we observe that the only alphabetical symbol representing an unknown quantity is $x$. There are no secondary variables (such as $y$ or $t$) present in any of the terms.

Conclusion for Condition 1: The expression is strictly in one variable.

Step 2: Analyzing the Exponents of Each Term

To rigorously verify Condition 2, we must decompose the expression into its constituent terms and isolate the exponent of $x$ for each. Note that constants can be expressed as coefficients of $x^0$ [By the Zero Exponent Rule: $x^0 = 1$ for $x \neq 0$].

Step 3: Visualizing the Polynomial Structure

The following diagram maps the anatomical structure of the given expression, proving that all exponents belong to the set of whole numbers and only a single variable is utilized.

Step 4: Synthesizing the Findings

Because the expression contains exactly one variable ($x$) and every exponent of $x$ across all terms is a non-negative integer ($2, 1, \text{ and } 0$), the expression perfectly satisfies all algebraic axioms required to be classified as a polynomial in one variable.

Final Solution: The expression $4x^2 - 3x + 7$ is a polynomial in one variable. The reason is that it contains only a single variable ($x$), and the exponents of the variable in all terms ($2, 1, \text{ and } 0$) are whole numbers.

Initial Setup & Given Expression

We are tasked with classifying the following algebraic expression based on its degree:

$P(y) = y + y^2 + 4$

Step 1: Rearranging into Standard Form

By mathematical convention, polynomials are typically written in standard form, where the terms are ordered from the highest power of the variable to the lowest power [Descending Order of Exponents]. This makes it easier to identify the leading term and the degree.

Rearranging the terms of $P(y)$ according to the powers of $y$:

$P(y) = y^2 + y + 4$

Step 2: Analyzing the Exponents of Each Term

To classify the polynomial, we must determine the exponent of the variable $y$ in each distinct term. We apply the fundamental laws of exponents to reveal hidden powers:

• First term ($y^2$): The exponent of the variable $y$ is explicitly $2$.
• Second term ($y$): Any variable written without an explicit exponent has an implied exponent of $1$. Thus, $y = y^1$. The exponent is $1$.
• Third term ($4$): This is a constant term. By the zero exponent rule [where $y^0 = 1$ for $y \neq 0$], a constant $c$ can be written as $c \cdot y^0$. Therefore, $4 = 4y^0$. The exponent of $y$ here is $0$.

Step 3: Determining the Degree of the Polynomial

The degree of a polynomial in one variable is defined as the highest power (maximum exponent) of the variable present in the expression, provided its coefficient is non-zero.

Comparing the set of exponents we identified $\{2, 1, 0\}$, the maximum value is $2$.

Therefore, the degree of the polynomial $P(y)$ is $2$.

Step 4: Classification Based on Degree

In algebraic nomenclature, polynomials are classified strictly according to their degree. The standard classifications are outlined in the table below:

Because the highest exponent of the variable $y$ in the expression $y^2 + y + 4$ is exactly $2$, it perfectly matches the definition of a quadratic polynomial.

Final Solution

Final Solution: The given polynomial $y + y^2 + 4$ has a degree of 2, and is therefore classified as a quadratic polynomial.

Given Variables & Initial Setup

Let the given algebraic expression be denoted as a polynomial function $P(x)$:

$P(x) = 2 + x^2 + x$

Our objective is to determine the coefficient of the $x^2$ term. [A coefficient is defined mathematically as the numerical or constant multiplier of a specific variable within an algebraic term].

Step 1: Standardizing the Polynomial Form

To systematically analyze the polynomial, we first rearrange its terms in descending order of their degree to express it in standard form.

$P(x) = x^2 + x + 2$

[Per the standard convention of polynomial representation, where terms are ordered from the highest exponent to the lowest exponent].

Step 2: Isolating the Target Term

We inspect the standardized polynomial to locate the specific term containing the variable $x$ raised to the power of $2$. The polynomial consists of three distinct terms separated by addition:

• First term: $x^2$ (Quadratic term, Degree 2)
• Second term: $x$ (Linear term, Degree 1)
• Third term: $2$ (Constant term, Degree 0)

The target term for our analysis is $+x^2$.

Step 3: Extracting the Coefficient

We must identify the constant numerical value that multiplies the variable $x^2$. In algebra, when a variable does not have an explicitly written numerical prefix, it is implicitly multiplied by $1$.

$x^2 = 1 \cdot x^2$

[By the Multiplicative Identity Property of Real Numbers, which states that any real number or variable multiplied by $1$ remains unchanged, i.e., $a \cdot 1 = a$].

Therefore, the numerical multiplier (coefficient) of the $x^2$ term is exactly $1$.

Final Solution: The coefficient of $x^2$ in the polynomial $2 + x^2 + x$ is $1$.

Given Variables & Initial Setup

We are given the algebraic expression:

$P(x) = x - x^3$

Our objective is to classify this polynomial into one of three categories: linear, quadratic, or cubic. To do this, we must determine the degree of the polynomial.

Step 1: Expressing the Polynomial in Standard Form

The standard form of a polynomial in one variable requires arranging the terms in descending order of their exponents [Per the conventions of algebraic notation].

The given polynomial has two terms:

• The term $x$, which can be written as $x^1$.
• The term $-x^3$.

Rearranging these terms in descending order of their powers, we get:

$P(x) = -x^3 + x^1$

Step 2: Determining the Degree of the Polynomial

The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable that has a non-zero coefficient.

Analyzing the exponents in our standard form polynomial $P(x) = -x^3 + x^1$:

• The exponent of the first term ($-x^3$) is $3$.
• The exponent of the second term ($x^1$) is $1$.

Comparing the exponents, $3 > 1$. Therefore, the highest power of the variable $x$ is $3$.

$ \text{Degree of } P(x) = 3 $

Step 3: Classifying the Polynomial

Polynomials are classified based on their degree according to the following definitions:

Since the degree of $P(x) = -x^3 + x$ is exactly $3$, it satisfies the definition of a cubic polynomial [Where $a = -1, b = 0, c = 1, d = 0$].

Visual Verification: Graphical Representation of a Cubic Polynomial

A cubic polynomial typically exhibits a characteristic "S-shaped" curve and can intersect the x-axis up to three times [Per the Fundamental Theorem of Algebra]. Below is the precise geometric plot of $y = x - x^3$, clearly showing its three real roots at $x = -1$, $x = 0$, and $x = 1$.

Final Solution: The highest power of the variable $x$ in the polynomial $x - x^3$ is $3$. Therefore, it is classified as a cubic polynomial.

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(t) = 5t - \sqrt{7}$

The degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the polynomial with a non-zero coefficient. [Per the Fundamental Theorem of Algebra and standard polynomial definitions]. To find the degree, we must analyze the exponent of the variable $t$ in every term of the expression.

Step 1: Isolate and Analyze Each Term

A polynomial is constructed as a sum of individual terms. Let us break down the given polynomial $P(t)$ into its constituent terms:

• Term 1: $5t$
• Term 2: $-\sqrt{7}$

Step 2: Determine the Exponent of the Variable in Each Term

We rewrite each term to explicitly reveal the power of the variable $t$:

• Analyzing Term 1 ($5t$): The variable $t$ is written without an explicit exponent. By algebraic convention, any variable without a written exponent has an implicit exponent of $1$. Thus, it can be written as $5t^1$. The power of the variable in this term is $1$.
• Analyzing Term 2 ($-\sqrt{7}$): This is a constant term and contains no visible variable. By the laws of exponents, any non-zero variable raised to the power of $0$ equals $1$ (i.e., $t^0 = 1$). Thus, the constant $-\sqrt{7}$ can be mathematically expressed as $-\sqrt{7}t^0$. The power of the variable in this term is $0$.

Step 3: Identify the Highest Power (Degree)

We now compare the exponents of $t$ extracted from all terms:

• Power in Term 1: $1$
• Power in Term 2: $0$

The degree is the maximum value among these exponents: $\max(1, 0) = 1$. Therefore, the highest power of the variable $t$ is $1$.

Visual Representation: The Linear Nature of Degree 1 Polynomials

A polynomial of degree $1$ is classified geometrically as a linear polynomial. When graphed on a Cartesian coordinate system, a degree $1$ polynomial always forms a perfectly straight line. Below is the exact geometric representation of $P(t) = 5t - \sqrt{7}$, demonstrating its constant rate of change (slope = $5$) and its intercepts.

Final Solution: The degree of the polynomial $5t - \sqrt{7}$ is $1$.

Given Polynomial & Initial Setup

We are tasked with classifying the following algebraic expression based on its degree:

$P(x) = x^2 + x$

Step 1: Theoretical Foundation of Polynomial Classification

In algebra, polynomials in a single variable are classified according to their degree. [Per the Fundamental Theorem of Algebra and polynomial definitions], the degree of a polynomial is defined as the highest exponent (power) of the variable present in the expression with a non-zero coefficient.

• Linear Polynomial: A polynomial of degree $1$. Standard form: $ax + b$ (where $a \neq 0$).
• Quadratic Polynomial: A polynomial of degree $2$. Standard form: $ax^2 + bx + c$ (where $a \neq 0$).
• Cubic Polynomial: A polynomial of degree $3$. Standard form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).

Step 2: Term-by-Term Analysis of $P(x)$

We decompose the given polynomial $P(x) = x^2 + x$ into its constituent terms to identify the exponents of the variable $x$:

• First Term ($x^2$): The variable $x$ is raised to the power of $2$.
• Second Term ($x$): The variable $x$ is implicitly raised to the power of $1$ (since $x = x^1$).

Comparing the exponents $\{2, 1\}$, the maximum value is $2$. Therefore, the highest power of the variable $x$ in the polynomial is $2$.

Step 3: Classification and Geometric Justification

Since the highest exponent is $2$, the degree of the polynomial $P(x) = x^2 + x$ is exactly $2$. By definition, a polynomial of degree $2$ is classified as a quadratic polynomial.

[Geometrically, a quadratic polynomial represents a parabola when graphed on a Cartesian plane. The presence of the $x^2$ term dictates this parabolic curvature, distinguishing it from the straight line of a linear polynomial or the inflection curve of a cubic polynomial.]

Final Solution: The highest power of the variable $x$ in the expression $x^2 + x$ is $2$. Therefore, it is classified as a quadratic polynomial.

Initial Setup & Polynomial Definition

We are given the following algebraic expression in terms of the variable $y$:

$P(y) = 4 - y^2$

[Per the fundamental theorem of algebra and polynomial definitions], a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Step 1: Standard Form Representation

To rigorously analyze the polynomial, we first rewrite it in its standard form. The standard form of a single-variable polynomial dictates that the terms must be ordered in descending powers of the variable.

$P(y) = -y^2 + 4$

Next, we express every term explicitly with the variable $y$ to expose all hidden exponents. A constant term, such as $4$, can be written as $4y^0$ [since $y^0 = 1$ for all $y \neq 0$].

$P(y) = -1 \cdot y^2 + 4 \cdot y^0$

Step 2: Exponent Analysis & Term Breakdown

We now isolate and evaluate the exponent of the variable $y$ in each distinct term of the polynomial.

Step 3: Determination of the Degree

The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable that appears with a non-zero coefficient.

• The exponent of the first term is $2$.
• The exponent of the second term is $0$.

Comparing the exponents: $\max(2, 0) = 2$.

Because the highest exponent is $2$, the polynomial is classified as a quadratic polynomial.

Geometric Visualization: The Quadratic Curve

To provide a comprehensive understanding, we can visualize the polynomial by plotting the relation $x = 4 - y^2$. Because the degree is $2$, the geometric representation is a parabola. Since the squared variable is $y$ and its coefficient is negative, the parabola opens to the left, with its vertex at the maximum $x$-value.

Final Solution: The degree of the polynomial $4 - y^2$ is 2.

Initial Setup & Theoretical Foundation

We are given the mathematical expression:

$P(x) = 3$

To determine the degree of this polynomial, we must first establish the formal definition of a polynomial's degree. [Per the Fundamental Theorem of Algebra and standard polynomial theory], the degree of a polynomial in a single variable is defined as the highest exponent (power) of the variable that possesses a non-zero coefficient.

Step 1: Algebraic Manipulation of the Constant

The given expression is a constant number, $3$. In algebra, any non-zero constant can be expressed as a product of that constant and a variable raised to the power of zero.

Let us introduce a variable, $x$. [By the Zero Exponent Rule of indices, $x^0 = 1$ for all $x \neq 0$]. Therefore, we can rewrite the constant polynomial as follows:

$P(x) = 3 \cdot 1$

$P(x) = 3 \cdot x^0$

Step 2: Identification of the Highest Power

A polynomial in one variable $x$ is generally written in the standard descending form:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x^1 + a_0 x^0$

By mapping our expression $P(x) = 3x^0$ to the standard form, we observe:

• The coefficient $a_0 = 3$.
• All other coefficients ($a_1, a_2, \dots, a_n$) are exactly $0$.

Since the only term with a non-zero coefficient is $3x^0$, the highest power of the variable $x$ present in the polynomial is $0$.

Step 3: Graphical Verification (Geometric Interpretation)

Geometrically, the degree of a polynomial $P(x)$ correlates with the maximum number of times its graph can intersect the x-axis (its roots). The graph of $P(x) = 3$ is a horizontal line parallel to the x-axis, maintaining a constant y-value of $3$. Because it is parallel to the x-axis and not coincident with it, it never intersects the x-axis. Zero intersections imply zero roots, which perfectly aligns with a polynomial of degree $0$.

Theoretical Exception Note

It is critical to distinguish between a non-zero constant polynomial (like $3$) and the zero polynomial ($P(x) = 0$). While the degree of any non-zero constant polynomial is strictly $0$, the degree of the zero polynomial is mathematically undefined because $0$ can be written as $0x^0$, $0x^1$, $0x^{100}$, etc., making it impossible to define a unique highest power.

Final Solution: The degree of the polynomial $3$ is $0$.

Step 1: Initial Setup and Definition of the Polynomial

We are given the algebraic expression:

$P(x) = 7x^3$

To classify this mathematical expression, we must first verify that it satisfies the formal definition of a polynomial in one variable. A polynomial in a single variable $x$ is an expression consisting of variables and coefficients, where the exponents of the variables are non-negative integers.

• Coefficient: $7$ (a real number)
• Variable: $x$
• Exponent: $3$ (a non-negative integer)

Since the exponent is a non-negative integer, $P(x) = 7x^3$ is strictly a polynomial.

Step 2: Determining the Degree of the Polynomial

The classification of a polynomial is fundamentally determined by its degree. [By definition, the degree of a polynomial in one variable is the highest power (exponent) of the variable present in the expression with a non-zero coefficient].

Analyzing our given polynomial:

$P(x) = 7x^3$

This is a monomial (a polynomial with a single term). The only variable present is $x$, and its exponent is $3$. Therefore, the highest power of $x$ in this polynomial is $3$.

$ \text{Degree of } P(x) = 3 $

Step 3: Classifying the Polynomial

Polynomials are classified by their degree according to the following standard algebraic nomenclature:

Because the degree of $7x^3$ is exactly $3$, it falls into the category of a cubic polynomial.

Step 4: Geometric Interpretation (Visualizing the Cubic Polynomial)

A cubic polynomial of the form $P(x) = ax^3$ (where $a > 0$) produces a characteristic curve that passes through the origin $(0,0)$ and exhibits point symmetry about the origin. This is known as an inflection point, where the concavity of the graph changes. Below is the precise geometric representation of $y = 7x^3$.

Final Solution: The highest power of the variable $x$ in the expression $7x^3$ is $3$. Therefore, $7x^3$ is classified as a cubic polynomial.

Initial Setup & Given Expression

We are given the following algebraic expression:

$P(r) = r^2$

Our objective is to classify this polynomial as linear, quadratic, or cubic based on its mathematical properties.

Step 1: Theoretical Foundation of Polynomial Classification

In algebra, polynomials are classified according to their degree. The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable present in the expression with a non-zero coefficient. [Per the Fundamental Theorem of Algebra and standard polynomial definitions].

The standard classifications based on degree are as follows:

• Linear Polynomial: A polynomial of degree $1$. General form: $ax + b$ (where $a \neq 0$).
• Quadratic Polynomial: A polynomial of degree $2$. General form: $ax^2 + bx + c$ (where $a \neq 0$).
• Cubic Polynomial: A polynomial of degree $3$. General form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).

Step 2: Analyzing the Degree of the Given Polynomial

Let us examine the given polynomial:

$P(r) = r^2$

This expression is a monomial (a polynomial consisting of a single term). We analyze the components of this term:

• The variable is $r$.
• The coefficient is $1$ (since $r^2$ is equivalent to $1 \cdot r^2$).
• The exponent of the variable $r$ is $2$.

Because there are no other terms in the polynomial, the highest power of the variable $r$ is exactly $2$. Therefore, the degree of the polynomial $P(r) = r^2$ is $2$.

Step 3: Visualizing the Quadratic Nature

To rigorously confirm the nature of this polynomial, we can observe its graphical representation. A polynomial of degree $2$ forms a parabola when plotted on a Cartesian coordinate system. The precise geometric mapping of $P(r) = r^2$ demonstrates a non-linear, symmetric curve with a single vertex at the origin $(0,0)$.

Step 4: Final Classification

Because the highest exponent of the variable $r$ is $2$, the polynomial satisfies the strict definition of a quadratic polynomial. It does not possess a degree of $1$ (which would make it linear) nor a degree of $3$ (which would make it cubic).

Final Solution: The polynomial $r^2$ is a quadratic polynomial.

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(x) = \sqrt{2}x - 1$

Our objective is to determine the coefficient of the $x^2$ term within this polynomial. [By definition, a coefficient is the numerical multiplier of a variable in a specific term of an algebraic expression].

Step 1: Analyzing the Degree and Terms of the Polynomial

The given polynomial, $P(x) = \sqrt{2}x - 1$, consists of two explicitly written terms:

• The linear term: $\sqrt{2}x$, where the variable $x$ is raised to the power of $1$. The coefficient here is $\sqrt{2}$.
• The constant term: $-1$, which can be mathematically expressed as $-1x^0$.

Because the highest power of the variable $x$ present in the expression is $1$, this is classified as a linear polynomial (degree $1$).

Step 2: Expressing the Polynomial in Standard Form

To rigorously identify the coefficient of a term that does not explicitly appear in the expression (in this case, the $x^2$ term), we must rewrite the polynomial in its complete standard descending form. [Per the axioms of polynomial representation, any missing term of degree $k$ can be introduced into the expression by multiplying it by zero ($0 \cdot x^k$), because adding the additive identity ($0$) does not alter the intrinsic value of the polynomial].

We expand $P(x)$ to explicitly include the quadratic ($x^2$) term:

$P(x) = 0 \cdot x^2 + \sqrt{2}x - 1$

Step 3: Visualizing the Polynomial Structure

The diagram below illustrates the structural breakdown of the polynomial when expanded to include the quadratic term, isolating the coefficient for each degree of $x$.

Step 4: Extracting the Target Coefficient

By comparing our expanded polynomial $P(x) = 0x^2 + \sqrt{2}x - 1$ with the general quadratic form $ax^2 + bx + c$, we can map the coefficients directly:

• $a$ (coefficient of $x^2$) = $0$
• $b$ (coefficient of $x$) = $\sqrt{2}$
• $c$ (constant term) = $-1$

Since the $x^2$ term is entirely absent from the original expression, its numerical multiplier must be zero.

Final Solution: The coefficient of $x^2$ in the polynomial $\sqrt{2}x - 1$ is $0$.

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(x) = \frac{\pi}{2}x^2 + x$

In algebra, a polynomial is defined as an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. [Per the Fundamental Theorem of Algebra and standard polynomial definitions], a coefficient is the numerical or constant multiplier attached to a specific power of a variable within a term.

Step 1: Term-by-Term Analysis of the Polynomial

To rigorously identify the coefficient, we must first decompose the polynomial $P(x)$ into its constituent terms. The terms of a polynomial are separated by addition ($+$) or subtraction ($-$) operators.

• Term 1: $\frac{\pi}{2}x^2$ (This is the quadratic term, possessing a degree of 2).
• Term 2: $x$ (This is the linear term, possessing a degree of 1, which can be written as $1 \cdot x^1$).

Step 2: Isolating the Target Variable

The objective is to find the coefficient specifically for the $x^2$ variable. We isolate the term containing $x^2$, which is:

$\frac{\pi}{2}x^2$

Step 3: Extraction and Classification of the Coefficient

By observing the isolated term $\frac{\pi}{2}x^2$, we separate the variable component ($x^2$) from its constant multiplier.

The constant multiplier is $\frac{\pi}{2}$.

Analytical Note: It is crucial to recognize that polynomials over the real numbers ($\mathbb{R}$) can have irrational coefficients. The number $\pi$ is an irrational mathematical constant (approximately $3.14159...$). Therefore, the fraction $\frac{\pi}{2}$ is a perfectly valid real number and serves as the exact coefficient of the $x^2$ term.

Final Solution: The coefficient of $x^2$ in the polynomial $\frac{\pi}{2}x^2 + x$ is $\frac{\pi}{2}$.

Step 1: Initial Setup & Algebraic Transformation

The given algebraic expression is:

$y + \frac{2}{y}$

To rigorously analyze whether this expression is a polynomial, we must first express all terms such that the variable appears in the numerator. [Per the Laws of Exponents, specifically the negative exponent rule $\frac{1}{a^n} = a^{-n}$], we rewrite the second term of the expression:

$y^1 + 2y^{-1}$

Step 2: Variable Analysis

The problem requires us to verify two distinct conditions: whether the expression is in one variable, and whether it is a polynomial.

Looking at the transformed expression $y^1 + 2y^{-1}$, the only alphabetical symbol representing an unknown quantity is $y$. There are no other variables (such as $x$, $z$, etc.) present. Therefore, the expression strictly satisfies the condition of being in one variable.

Step 3: Theoretical Definition of a Polynomial

By mathematical definition, an algebraic expression is classified as a polynomial if and only if it can be written in the form:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

Where:

• $a_n, a_{n-1}, \dots, a_0$ are real number coefficients.
• The exponents $n, n-1, \dots$ must belong strictly to the set of non-negative integers (i.e., the set of whole numbers, $\mathbb{W} = \{0, 1, 2, 3, \dots\}$).

Step 4: Exponent Analysis

We now evaluate the exponents of the variable $y$ in our transformed expression $y^1 + 2y^{-1}$ against the theoretical definition:

• First Term ($y^1$): The exponent of $y$ is $1$. Because $1 \in \mathbb{W}$, this specific term satisfies the polynomial condition.
• Second Term ($2y^{-1}$): The exponent of $y$ is $-1$. Because $-1 \notin \mathbb{W}$ (it is a negative integer, not a whole number), this term violates the fundamental definition of a polynomial.

Step 5: Logical Conclusion

For an entire expression to be classified as a polynomial, every single term must possess a whole number exponent. The presence of even one term with a negative or fractional exponent immediately disqualifies the entire expression.

Final Solution: The expression $y + \frac{2}{y}$ is NOT a polynomial. While it is an expression in exactly one variable ($y$), the exponent of the variable in the second term is $-1$, which is not a whole number.

Step 1: Theoretical Foundation of Polynomials

To determine whether a given algebraic expression is a polynomial in one variable, we must evaluate it against two strict algebraic criteria:

• Single Variable Condition: The expression must contain exactly one distinct variable (e.g., only $x$, only $y$, or only $z$).
• Non-Negative Integer Exponent Condition: [Per the fundamental definition of polynomials], the exponent of the variable in every term must be a whole number (i.e., $0, 1, 2, 3, \dots$). Fractional, negative, or irrational exponents on the variable disqualify the expression from being a polynomial.

Step 2: Structural Analysis of the Given Expression

The given mathematical expression is:

$y^2 + \sqrt{2}$

We can rewrite this expression in its standard canonical form by explicitly showing the variable in the constant term. Any non-zero constant $c$ can be written as $c \cdot y^0$ because $y^0 = 1$ (for $y \neq 0$).

$y^2 + \sqrt{2} \cdot y^0$

Step 3: Evaluating the Variable Condition

Observing the expression $y^2 + \sqrt{2}$, the only alphabetical symbol representing an unknown quantity is $y$. There are no other variables (such as $x$ or $z$) present in the expression. Therefore, the expression satisfies the condition of being in one variable.

Step 4: Evaluating the Exponent Condition

We must now inspect the exponent of the variable $y$ in each term:

Note: The presence of $\sqrt{2}$ does not violate the polynomial definition. The restriction of having non-negative integer powers applies only to the variables, not to the numerical coefficients or constants. $\sqrt{2}$ is simply a real number acting as the constant term.

Step 5: Logical Conclusion

Since the expression contains exactly one variable ($y$) and all exponents of this variable ($2$ and $0$) are non-negative integers (whole numbers), it perfectly satisfies all algebraic axioms defining a polynomial in one variable.

Final Solution: Yes, the expression $y^2 + \sqrt{2}$ is a polynomial in one variable. The reason is that it contains only one variable ($y$), and the exponent of the variable in every term is a whole number.

Given Polynomial & Initial Setup

We are given the following algebraic expression:

$P(t) = 3t$

Our objective is to classify this polynomial into one of three distinct algebraic categories: linear, quadratic, or cubic. The classification of a polynomial is strictly determined by its degree.

Step 1: Identify the Variable and its Exponents

In the given polynomial $P(t) = 3t$, the single independent variable present is $t$. To determine the nature of the polynomial, we must inspect the exponents attached to this variable.

The term $3t$ can be explicitly rewritten by revealing its hidden exponent:

$P(t) = 3t^1$

[Per the Fundamental Laws of Exponents, any variable $x$ written without a visible power is mathematically understood to have an exponent of $1$, i.e., $x = x^1$].

Step 2: Determine the Degree of the Polynomial

The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable in that expression.

• The only term containing the variable is $3t^1$.
• The exponent of $t$ in this term is $1$.

Therefore, the highest power of the variable $t$ is $1$. This establishes that the degree of the polynomial $P(t) = 3t$ is exactly $1$.

Step 3: Apply Polynomial Classification Criteria

Polynomials are universally classified based on their degree [By the Fundamental Theorem of Algebra and standard algebraic nomenclature]. The classification matrix is as follows:

Because the degree of $3t$ is $1$, it perfectly satisfies the condition for a linear polynomial. It matches the standard linear form $at + b$ where the leading coefficient $a = 3$ and the constant term $b = 0$.

Step 4: Geometric Verification (Visualizing Linearity)

A defining characteristic of a linear polynomial is that its graph forms a perfectly straight line [Per Cartesian Coordinate Geometry]. Below is the precise geometric representation of $P(t) = 3t$, demonstrating a constant rate of change (slope $m = 3$).

Final Solution: The given polynomial $3t$ has a highest exponent (degree) of $1$. Therefore, it is classified as a linear polynomial.

Given Polynomial & Initial Setup

We are given the algebraic expression:

$P(x) = 2 - x^2 + x^3$

Our objective is to determine the numerical coefficient of the $x^2$ term within this polynomial.

Step 1: Understanding Polynomial Terms and Standard Form

[Per the algebraic definition of a polynomial], an expression is composed of distinct terms separated by addition ($+$) or subtraction ($-$) operators. To analyze the polynomial systematically, it is best practice to rewrite it in standard form, where the terms are ordered by descending powers of the variable $x$.

Rearranging $P(x)$ yields:

$P(x) = x^3 - x^2 + 2$

Step 2: Isolating the Target Term

We must identify the specific term that contains the variable $x^2$. By dissecting the polynomial into its constituent terms, we get:

• Cubic term: $+x^3$
• Quadratic term: $-x^2$
• Constant term: $+2$

The target term containing $x^2$ is explicitly $-x^2$.

Step 3: Extracting the Numerical Coefficient

[By the fundamental axioms of algebra], the coefficient is the constant multiplicative factor attached to a variable in a specific term. When a variable appears without an explicit numerical multiplier but carries a negative sign, the multiplicative identity property dictates that the hidden multiplier is $-1$.

We can expand the target term algebraically to reveal its coefficient:

$-x^2 = (-1) \cdot x^2$

Thus, the numerical factor multiplying $x^2$ is $-1$.

Visual Breakdown of the Polynomial Terms

Final Solution: The coefficient of $x^2$ in the polynomial $2 - x^2 + x^3$ is $-1$.

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(t) = 3\sqrt{t} + t\sqrt{2}$

To determine whether this expression is a polynomial in one variable, we must evaluate it against the fundamental definition of a polynomial. [Per the algebraic definition of polynomials], an expression is classified as a polynomial in one variable if and only if:

• It contains exactly one variable.
• The exponent of the variable in every term is a non-negative integer (i.e., a whole number belonging to the set $\mathbb{W} = \{0, 1, 2, 3, \dots\}$).
• The coefficients of the variables are real numbers ($\mathbb{R}$).

Step 1: Algebraic Transformation of Radical Terms

To properly analyze the exponents, we must convert all radical notations into fractional exponents using the laws of indices. [By the definition of rational exponents, $\sqrt[n]{x^m} = x^{m/n}$].

Applying this to the given expression:

• The first term is $3\sqrt{t}$. The square root of $t$ can be rewritten as $t^{1/2}$. Thus, the term becomes $3t^{1/2}$.
• The second term is $t\sqrt{2}$. The variable $t$ is raised to the first power, and the coefficient is $\sqrt{2}$. This can be rewritten as $\sqrt{2}t^1$.

The transformed expression is:

$P(t) = 3t^{1/2} + \sqrt{2}t^1$

Step 2: Exponent Analysis & Verification

We now isolate and examine the exponent of the variable $t$ in each term of the transformed expression.

Step 3: Logical Deduction

While the expression contains only one variable ($t$), the exponent of $t$ in the first term is $\frac{1}{2}$. Because $\frac{1}{2}$ is a rational number but not a non-negative integer (whole number), the expression violates the strict algebraic criteria required for polynomials.

Final Solution: The expression $3\sqrt{t} + t\sqrt{2}$ is NOT a polynomial in one variable. Reason: The exponent of the variable $t$ in the first term is $\frac{1}{2}$, which is not a whole number.

Initial Setup & Given Expression

We are given the algebraic expression:

$P(x) = 1 + x$

Our objective is to classify this polynomial into one of three categories: linear, quadratic, or cubic. This classification is strictly determined by the degree of the polynomial.

Step 1: Standardizing the Polynomial

First, we rewrite the polynomial in its standard form, where the terms are ordered in descending powers of the variable $x$.

$P(x) = x + 1$

We can explicitly write the exponents for every term to ensure absolute clarity:

• The term $x$ is equivalent to $x^1$.
• The constant term $1$ is equivalent to $1 \cdot x^0$ [Per the Zero Exponent Rule, where $x^0 = 1$ for $x \neq 0$].

Thus, the fully expanded standard form is:

$P(x) = 1 \cdot x^1 + 1 \cdot x^0$

Step 2: Determining the Degree of the Polynomial

The degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the expression with a non-zero coefficient.

Analyzing the exponents in $P(x) = x^1 + 1 \cdot x^0$:

• The exponent of the first term is $1$.
• The exponent of the second term is $0$.

Comparing these values, the maximum exponent is $1$. Therefore, the degree of the polynomial $P(x) = 1 + x$ is exactly $1$.

Step 3: Applying the Classification Theorem

Polynomials are classified by their degree according to the following universally accepted algebraic definitions:

Since we have rigorously determined that the degree of $1 + x$ is $1$, it maps directly to the definition of a linear polynomial.

Visual Verification: Geometric Interpretation

A linear polynomial, when graphed on a Cartesian coordinate system as a function $y = P(x)$, will always produce a perfectly straight line [Per the geometric definition of first-degree equations]. Below is the precise graphical representation of $y = x + 1$, demonstrating its constant slope ($m = 1$) and y-intercept ($c = 1$).

Final Solution: The polynomial $1 + x$ has a highest degree of $1$. Therefore, it is classified as a linear polynomial.