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Q4(iii):
Find the zero of the polynomial in each of the following cases:
(iii) $p(x) = 2x + 5$
Solution :
Given Variables & Initial Setup
We are given the linear polynomial:
$p(x) = 2x + 5$
In algebra, the zero of a polynomial is defined as the specific value of the variable (in this case, $x$) that makes the value of the entire polynomial equal to zero. Geometrically, this corresponds to the x-coordinate of the point where the graph of the polynomial intersects the x-axis (the x-intercept).
Step 1: Formulating the Equation
To find the zero of the polynomial, we must equate the polynomial function $p(x)$ to $0$. [Per the Fundamental Theorem of Algebra, a polynomial of degree 1 will have exactly one real zero].
$p(x) = 0$
Substituting the given expression for $p(x)$:
$2x + 5 = 0$
Step 2: Algebraic Isolation of the Variable
We must isolate $x$ using standard algebraic operations.
- Step 2a: Subtract $5$ from both sides of the equation to isolate the term containing $x$ [By the Subtraction Property of Equality].
$2x = -5$ - Step 2b: Divide both sides by $2$ to solve for $x$ [By the Division Property of Equality].
$x = \frac{-5}{2}$
This fraction can also be expressed as the decimal $-2.5$.
Step 3: Geometric Representation (Visual Proof)
Graphing the linear equation $y = 2x + 5$ provides a visual confirmation of the zero. The line crosses the x-axis exactly at $x = -2.5$.
Step 4: Verification
To ensure absolute mathematical rigor, we substitute $x = -\frac{5}{2}$ back into the original polynomial to verify that it yields $0$.
$p\left(-\frac{5}{2}\right) = 2\left(-\frac{5}{2}\right) + 5$
$p\left(-\frac{5}{2}\right) = -5 + 5$
$p\left(-\frac{5}{2}\right) = 0$
Since the result is exactly zero, the calculated value is verified as correct.
Final Solution: The zero of the polynomial $p(x) = 2x + 5$ is $x = -\frac{5}{2}$.
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(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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