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Q3(i):
Verify whether the following are zeroes of the polynomial, indicated against them.
(i) $p(x) = 3x + 1, x = –\frac{1}{3}$
Solution :
Given Variables & Initial Setup
We are given a linear polynomial in one variable, $x$, defined as:
$p(x) = 3x + 1$
We are tasked with verifying whether the specific value $x = -\frac{1}{3}$ is a zero of this polynomial.
[Per the foundational definition in polynomial algebra, a real number $a$ is considered a zero (or root) of a polynomial $p(x)$ if and only if evaluating the polynomial at $x = a$ yields a result of zero. Mathematically, this is expressed as $p(a) = 0$.]
Step 1: Substitution of the Given Value
To test the condition $p(a) = 0$, we substitute $x = -\frac{1}{3}$ into the polynomial equation $p(x)$.
$p\left(-\frac{1}{3}\right) = 3\left(-\frac{1}{3}\right) + 1$
Step 2: Algebraic Evaluation
Next, we perform the multiplication operation. [By the properties of rational numbers, multiplying an integer by a fraction involves multiplying the integer by the numerator and dividing by the denominator].
$3 \times \left(-\frac{1}{3}\right) = \frac{3 \times -1}{3} = -1$
Substituting this product back into the polynomial expression:
$p\left(-\frac{1}{3}\right) = -1 + 1$
Step 3: Final Simplification and Logical Conclusion
Evaluating the arithmetic sum:
$p\left(-\frac{1}{3}\right) = 0$
Because the polynomial evaluates exactly to $0$, the condition for $x = -\frac{1}{3}$ being a zero of the polynomial is strictly satisfied.
Graphical Verification (Geometric Interpretation)
Geometrically, the zero of a linear polynomial $p(x) = mx + c$ corresponds to the x-intercept of the line $y = mx + c$ on a Cartesian coordinate plane. The graph below demonstrates the line $y = 3x + 1$ intersecting the x-axis precisely at the coordinate $\left(-\frac{1}{3}, 0\right)$.
Final Solution: Yes, $x = -\frac{1}{3}$ is a zero of the polynomial $p(x) = 3x + 1$ because $p\left(-\frac{1}{3}\right) = 0$.
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(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(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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