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Q3(iv):
Verify whether the following are zeroes of the polynomial, indicated against them.
(iv) $p(x) = (x + 1) (x – 2), x = – 1, 2$
Solution :
Initial Setup & Theoretical Foundation
We are given the polynomial function:
$p(x) = (x + 1)(x - 2)$
We must verify whether the given values, $x = -1$ and $x = 2$, are zeroes of the polynomial.
[By the Fundamental Definition of a Zero of a Polynomial: A real number $a$ is a zero of a polynomial $p(x)$ if and only if evaluating the polynomial at $x = a$ yields zero, i.e., $p(a) = 0$.]
Step 1: Evaluating the Polynomial at $x = -1$
To determine if $x = -1$ is a zero, we substitute $-1$ for every instance of $x$ in the polynomial $p(x)$.
- Substitute $x = -1$:
$p(-1) = (-1 + 1)(-1 - 2)$ - Simplify the terms within the parentheses:
$p(-1) = (0)(-3)$ - Apply the Zero Property of Multiplication [Any number multiplied by zero equals zero]:
$p(-1) = 0$
Since $p(-1) = 0$, we mathematically confirm that $x = -1$ is a zero of the polynomial $p(x)$.
Step 2: Evaluating the Polynomial at $x = 2$
Next, we apply the same substitution methodology for $x = 2$.
- Substitute $x = 2$:
$p(2) = (2 + 1)(2 - 2)$ - Simplify the terms within the parentheses:
$p(2) = (3)(0)$ - Apply the Zero Property of Multiplication:
$p(2) = 0$
Since $p(2) = 0$, we mathematically confirm that $x = 2$ is also a zero of the polynomial $p(x)$.
Graphical Verification
The polynomial $p(x) = (x + 1)(x - 2)$ expands to the quadratic form $p(x) = x^2 - x - 2$. Geometrically, the zeroes of a polynomial correspond to the $x$-intercepts of its graph. The precise Cartesian plot below demonstrates the parabola intersecting the $x$-axis exactly at the coordinates $(-1, 0)$ and $(2, 0)$.
Final Solution: Yes, both $x = -1$ and $x = 2$ are zeroes of the polynomial $p(x) = (x + 1)(x - 2)$, as evaluating the polynomial at both values yields exactly $0$.
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(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(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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