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Q4(ii):
Find the zero of the polynomial in each of the following cases:
(ii) $p(x) = x β 5$
Solution :
Given Polynomial & Theoretical Foundation
We are tasked with finding the zero of the following linear polynomial:
$p(x) = x - 5$
Step 1: Applying the Definition of a Zero
The zero (or root) of a polynomial $p(x)$ is defined as the specific real value of the variable $x$ for which the polynomial evaluates to zero. [Per the fundamental definition of polynomial roots]. Therefore, to find the zero, we must set the polynomial equal to zero:
$p(x) = 0$
Step 2: Algebraic Substitution and Manipulation
Substituting the given algebraic expression for $p(x)$ into our equation, we obtain:
$x - 5 = 0$
To isolate the variable $x$ and solve the linear equation, we add $5$ to both sides of the equation [Per the Addition Property of Equality, which states that adding the same number to both sides of an equation maintains its balance]:
$x - 5 + 5 = 0 + 5$
$x = 5$
Step 3: Rigorous Verification
To ensure absolute mathematical accuracy, we verify the solution by substituting $x = 5$ back into the original polynomial to check if it yields zero:
$p(5) = (5) - 5$
$p(5) = 0$
Since the evaluation results in exactly zero, $x = 5$ is confirmed as the correct zero of the polynomial.
Step 4: Graphical Interpretation
Geometrically, the zero of a polynomial with real coefficients corresponds to the $x$-coordinate of the point where the graph of the function $y = p(x)$ intersects the $x$-axis (where $y = 0$). For the linear function $y = x - 5$, the $x$-intercept occurs exactly at the Cartesian coordinate $(5, 0)$.
Final Solution: The zero of the polynomial $p(x) = x - 5$ is $x = 5$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the value of the polynomial $5x β 4x^2 + 3$ at (i) $x = 0$
- Q1(ii): Find the value of the polynomial $5x β 4x^2 + 3$ at (ii) $x = β1$
- Q1(iii): Find the value of the polynomial $5x β 4x^2 + 3$ at (iii) $x = 2$
- Q2(i): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (i) $p(y) = y^2 β y + 1$
- Q2(ii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (ii) $p(t) = 2 + t + 2t^2 β t^3$
- 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) $p(x) = (x β 1) (x + 1)$
- Q3(i): Verify whether the following are zeroes of the polynomial, indicated against them. (i) $p(x) = 3x + 1, x = β\frac{1}{3}$
- Q3(ii): Verify whether the following are zeroes of the polynomial, indicated against them. (ii) $p(x) = 5x β \pi, x = \frac{4}{5}$
- Q3(iii): Verify whether the following are zeroes of the polynomial, indicated against them. (iii) $p(x) = x^2 β 1, x = 1, β1$
- Q3(iv): Verify whether the following are zeroes of the polynomial, indicated against them. (iv) $p(x) = (x + 1) (x β 2), x = β 1, 2$
- 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) $p(x) = lx + m, x = β\frac{m}{l}$
- Q3(vii): Verify whether the following are zeroes of the polynomial, indicated against them. (vii) $p(x) = 3x^2 β 1, x = β\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$
- 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(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) $p(x) = 3x β 2$
- Q4(v): Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$
- Q4(vi): Find the zero of the polynomial in each of the following cases: (vi) $p(x) = ax, a \neq 0$
- 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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