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Q4(iv):
Find the zero of the polynomial in each of the following cases:
(iv) $p(x) = 3x – 2$
Solution :
Initial Setup & Theoretical Foundation
We are given the linear polynomial:
$p(x) = 3x - 2$
[Per the definition of the zero of a polynomial, a real number $a$ is a zero of a polynomial $p(x)$ if and only if $p(a) = 0$. Geometrically, this corresponds to the x-coordinate of the point where the graph of the polynomial intersects the x-axis.]
Step 1: Equating the Polynomial to Zero
To find the zero of the given polynomial, we must set the polynomial expression equal to zero. This transforms our polynomial into a linear equation.
$p(x) = 0$
$3x - 2 = 0$
Step 2: Algebraic Isolation of the Variable
We now solve for $x$ using standard algebraic manipulation [By the properties of equality].
- Addition Property of Equality: Add $2$ to both sides of the equation to isolate the term containing $x$.
$3x - 2 + 2 = 0 + 2$
$3x = 2$ - Division Property of Equality: Divide both sides by $3$ to solve for $x$.
$\frac{3x}{3} = \frac{2}{3}$
$x = \frac{2}{3}$
Step 3: Verification of the Zero
To ensure absolute mathematical rigor, we substitute $x = \frac{2}{3}$ back into the original polynomial to verify that it evaluates to zero.
$p\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right) - 2$
$p\left(\frac{2}{3}\right) = 2 - 2$
$p\left(\frac{2}{3}\right) = 0$
[Since the evaluation yields exactly zero, the calculated root is verified as correct.]
Graphical Representation of the Zero
The graph of the linear polynomial $y = 3x - 2$ is a straight line. The zero of the polynomial is the exact point where this line crosses the x-axis (where $y = 0$). As calculated, this intersection occurs at the coordinate $\left(\frac{2}{3}, 0\right)$.
Final Solution: The zero of the polynomial $p(x) = 3x - 2$ is $x = \frac{2}{3}$.
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(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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