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Q3(vi):
Verify whether the following are zeroes of the polynomial, indicated against them.
(vi) $p(x) = lx + m, x = –\frac{m}{l}$
Solution :
Given Variables & Initial Setup
We are given the linear polynomial:
$p(x) = lx + m$
We need to verify whether the given value of $x$ is a zero (or root) of the polynomial:
$x = -\frac{m}{l}$
[By the Fundamental Theorem of Algebra and the definition of polynomial roots, a real number $a$ is considered a "zero" of a polynomial $p(x)$ if and only if evaluating the polynomial at $x = a$ yields exactly zero, i.e., $p(a) = 0$. Furthermore, for $p(x)$ to be a valid linear polynomial of degree 1, the leading coefficient $l$ must not be equal to zero ($l \neq 0$).]
Step 1: Substitution of the Given Value
To test the condition $p(a) = 0$, we substitute $x = -\frac{m}{l}$ into the polynomial expression $p(x)$.
$p\left(-\frac{m}{l}\right) = l\left(-\frac{m}{l}\right) + m$
Step 2: Algebraic Simplification
Next, we perform the multiplication. Since $l$ is in the numerator of the coefficient and the denominator of the substituted fraction, they cancel each other out [Per the multiplicative inverse property, assuming $l \neq 0$]:
$p\left(-\frac{m}{l}\right) = \left(l \cdot \frac{-m}{l}\right) + m$
$p\left(-\frac{m}{l}\right) = -m + m$
Step 3: Final Evaluation
Combining the terms yields:
$p\left(-\frac{m}{l}\right) = 0$
Because the polynomial evaluates to zero at this specific value of $x$, the condition for it being a zero of the polynomial is perfectly satisfied.
Graphical Representation & Geometric Verification
Geometrically, the zero of a polynomial corresponds to the $x$-intercept of its graph. For the linear function $y = lx + m$, the line crosses the $x$-axis exactly at the coordinate $\left(-\frac{m}{l}, 0\right)$. The SVG below illustrates this relationship (assuming $l > 0$ and $m > 0$ for visual representation).
Final Solution: Yes, $x = -\frac{m}{l}$ is a zero of the polynomial $p(x) = lx + m$, because substituting this value into the polynomial 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(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(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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