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Q1(iv):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(iv) $4u^2 + 8u$
Solution :
Given: A quadratic polynomial $p(u) = 4u^2 + 8u$.
To Find: The zeroes of the polynomial and verify the relationship between the zeroes and the coefficients of the polynomial.
Step 1: Finding the zeroes of the polynomial
To find the zeroes of the polynomial $p(u)$, we set $p(u) = 0$.
$4u^2 + 8u = 0$
We factor out the greatest common factor, which is $4u$:
$4u(u + 2) = 0$ [Using the distributive property of multiplication over addition]
For the product to be zero, either $4u = 0$ or $u + 2 = 0$ [Zero Product Property].
Case 1: $4u = 0 \implies u = 0$
Case 2: $u + 2 = 0 \implies u = -2$
Thus, the zeroes of the polynomial are $\alpha = 0$ and $\beta = -2$.
Step 2: Identifying coefficients
Comparing the given polynomial $4u^2 + 8u$ with the standard form $au^2 + bu + c$, we have:
$a = 4$
$b = 8$
$c = 0$
Step 3: Verifying the relationship between zeroes and coefficients
The relationships to verify are:
1. Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2. Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Verification of Sum of Zeroes:
Sum of zeroes = $\alpha + \beta = 0 + (-2) = -2$
$-\frac{b}{a} = -\frac{8}{4} = -2$
Since $-2 = -2$, the relationship $\alpha + \beta = -\frac{b}{a}$ is verified.
Verification of Product of Zeroes:
Product of zeroes = $\alpha \cdot \beta = 0 \cdot (-2) = 0$
$\frac{c}{a} = \frac{0}{4} = 0$
Since $0 = 0$, the relationship $\alpha \cdot \beta = \frac{c}{a}$ is verified.
Final Answer: The zeroes of the polynomial $4u^2 + 8u$ are $0$ and $-2$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $-2$ and the product of zeroes is $0$.
More Questions from Class 10 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) $x^2 - 2x - 8$
- Q1(ii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (ii) $4s^2 - 4s + 1$
- Q1(iii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iii) $6x^2 - 3 - 7x$
- Q1(v): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (v) $t^2 - 15$
- Q1(vi): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (vi) $3x^2 - x - 4$
- Q2(i): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) $\frac{1}{4}, -1$
- Q2(ii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (ii) $\sqrt{2}, \frac{1}{3}$
- Q2(iii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iii) $0, \sqrt{5}$
- Q2(iv): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iv) $1, 1$
- Q2(v): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (v) $-\frac{1}{4}, \frac{1}{4}$
- Q2(vi): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (vi) $4, 1$
CBSE Solutions for Class 10 Mathematics Polynomials
Chapters in CBSE - Class 10 Mathematics
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