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Q4(vi):
Find the zero of the polynomial in each of the following cases:
(vi) $p(x) = ax, a \neq 0$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear polynomial:
$p(x) = ax$
with the strict mathematical constraint that $a \neq 0$.
Step 1: Formulating the Equation for the Zero
The "zero" (or root) of a polynomial is defined as the specific value of the independent variable $x$ that evaluates the polynomial to zero. [By the Fundamental Theorem of Algebra and the definition of polynomial roots].
To find this value, we must set the polynomial expression equal to zero:
$p(x) = 0$
$ax = 0$
Step 2: Algebraic Isolation of the Variable
To isolate $x$, we must divide both sides of the equation by the coefficient $a$. We are mathematically permitted to perform this operation strictly because the problem explicitly states the condition $a \neq 0$. [Per the Division Property of Equality, division by any non-zero real number is defined and preserves the equality].
$x = \frac{0}{a}$
Since zero divided by any non-zero number is strictly zero [Per the Zero Property of Division]:
$x = 0$
Step 3: Verification of the Root
To ensure absolute analytical accuracy, we substitute our derived root, $x = 0$, back into the original polynomial function:
$p(0) = a(0)$
$p(0) = 0$
Because the polynomial evaluates to $0$, the value $x = 0$ is rigorously verified as the correct zero of the polynomial.
Graphical Representation & Geometric Interpretation
Geometrically, the zero of a polynomial represents the exact x-coordinate where the graph of the function intersects the x-axis (where $p(x) = 0$). For any non-zero real value of $a$, the linear equation $p(x) = ax$ represents a straight line passing directly through the origin $(0,0)$. The steepness and direction of the line depend on $a$, but the x-intercept remains invariant at the origin.
Final Solution: The zero of the polynomial $p(x) = ax$ (where $a \neq 0$) is $x = 0$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the value of the polynomial $5x – 4x^2 + 3$ at (i) $x = 0$
- Q1(ii): Find the value of the polynomial $5x – 4x^2 + 3$ at (ii) $x = –1$
- Q1(iii): Find the value of the polynomial $5x – 4x^2 + 3$ at (iii) $x = 2$
- Q2(i): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (i) $p(y) = y^2 – y + 1$
- Q2(ii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (ii) $p(t) = 2 + t + 2t^2 – t^3$
- Q2(iii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iii) $p(x) = x^3$
- Q2(iv): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iv) $p(x) = (x – 1) (x + 1)$
- Q3(i): Verify whether the following are zeroes of the polynomial, indicated against them. (i) $p(x) = 3x + 1, x = –\frac{1}{3}$
- Q3(ii): Verify whether the following are zeroes of the polynomial, indicated against them. (ii) $p(x) = 5x – \pi, x = \frac{4}{5}$
- Q3(iii): Verify whether the following are zeroes of the polynomial, indicated against them. (iii) $p(x) = x^2 – 1, x = 1, –1$
- Q3(iv): Verify whether the following are zeroes of the polynomial, indicated against them. (iv) $p(x) = (x + 1) (x – 2), x = – 1, 2$
- Q3(v): Verify whether the following are zeroes of the polynomial, indicated against them. (v) $p(x) = x^2, x = 0$
- Q3(vi): Verify whether the following are zeroes of the polynomial, indicated against them. (vi) $p(x) = lx + m, x = –\frac{m}{l}$
- Q3(vii): Verify whether the following are zeroes of the polynomial, indicated against them. (vii) $p(x) = 3x^2 – 1, x = –\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$
- Q3(viii): Verify whether the following are zeroes of the polynomial, indicated against them. (viii) $p(x) = 2x + 1, x = \frac{1}{2}$
- Q4(i): Find the zero of the polynomial in each of the following cases: (i) $p(x) = x + 5$
- Q4(ii): Find the zero of the polynomial in each of the following cases: (ii) $p(x) = x – 5$
- Q4(iii): Find the zero of the polynomial in each of the following cases: (iii) $p(x) = 2x + 5$
- Q4(iv): Find the zero of the polynomial in each of the following cases: (iv) $p(x) = 3x – 2$
- Q4(v): Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$
- Q4(vii): Find the zero of the polynomial in each of the following cases: (vii) $p(x) = cx + d, c \neq 0, c, d$ are real numbers.
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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