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Q4(i):
Write the degree of each of the following polynomials:
(i) $5x^3 + 4x^2 + 7x$
Solution :
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 $3$.
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(ii): Write the degree of each of the following polynomials: (ii) $4 – y^2$
- Q4(iii): Write the degree of each of the following polynomials: (iii) $5t – \sqrt{7}$
- 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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