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Q4(iii):
Write the degree of each of the following polynomials:
(iii) $5t – \sqrt{7}$
Solution :
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 $1$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1
- Q1(i): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) $4x^2 – 3x + 7$
- Q1(ii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (ii) $y^2 + \sqrt{2}$
- Q1(iii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iii) $3\sqrt{t} + t\sqrt{2}$
- Q1(iv): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$
- Q1(v): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (v) $x^{10} + y^3 + t^{50}$
- Q2(i): Write the coefficients of $x^2$ in each of the following: (i) $2 + x^2 + x$
- Q2(ii): Write the coefficients of $x^2$ in each of the following: (ii) $2 – x^2 + x^3$
- Q2(iii): Write the coefficients of $x^2$ in each of the following: (iii) $\frac{\pi}{2}x^2 + x$
- Q2(iv): Write the coefficients of $x^2$ in each of the following: (iv) $\sqrt{2}x - 1$
- Q3: Give one example each of a binomial of degree 35, and of a monomial of degree 100.
- Q4(i): Write the degree of each of the following polynomials: (i) $5x^3 + 4x^2 + 7x$
- Q4(ii): Write the degree of each of the following polynomials: (ii) $4 – y^2$
- Q4(iv): Write the degree of each of the following polynomials: (iv) $3$
- Q5(i): Classify the following as linear, quadratic and cubic polynomials: (i) $x^2 + x$
- Q5(ii): Classify the following as linear, quadratic and cubic polynomials: (ii) $x – x^3$
- Q5(iii): Classify the following as linear, quadratic and cubic polynomials: (iii) $y + y^2 + 4$
- Q5(iv): Classify the following as linear, quadratic and cubic polynomials: (iv) $1 + x$
- Q5(v): Classify the following as linear, quadratic and cubic polynomials: (v) $3t$
- Q5(vi): Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$
- Q5(vii): Classify the following as linear, quadratic and cubic polynomials: (vii) $7x^3$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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