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Q1(i):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) $x^2 - 2x - 8$
Solution :
Given: A quadratic polynomial $p(x) = x^2 - 2x - 8$.
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 by factorization.
To find the zeroes of $p(x)$, we set $p(x) = 0$. Thus, we solve the quadratic equation:
$x^2 - 2x - 8 = 0$We use the splitting the middle term method. We look for two numbers whose product is the product of the coefficient of $x^2$ ($1$) and the constant term ($-8$), which is $-8$, and whose sum is the coefficient of $x$ ($-2$).
The factors of $-8$ are: $(1, -8), (-1, 8), (2, -4), (-2, 4)$.
The pair that sums to $-2$ is $(2, -4)$.
Rewriting the middle term:
$x^2 - 4x + 2x - 8 = 0$Grouping the terms:
$(x^2 - 4x) + (2x - 8) = 0$Factoring out the common terms from each group:
$x(x - 4) + 2(x - 4) = 0$ $(x - 4)(x + 2) = 0$Setting each factor to zero:
$x - 4 = 0 \implies x = 4$ $x + 2 = 0 \implies x = -2$Therefore, the zeroes of the polynomial are $\alpha = 4$ and $\beta = -2$.
Step 2: Verifying the relationship between zeroes and coefficients.
For a quadratic polynomial of the form $ax^2 + bx + c$, the relationships are:
1. Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2. Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Comparing $x^2 - 2x - 8$ with $ax^2 + bx + c$, we have:
$a = 1, b = -2, c = -8$Verification of Sum of Zeroes:
Sum of zeroes = $\alpha + \beta = 4 + (-2) = 2$
$-\frac{b}{a} = -\frac{-2}{1} = 2$
Since $2 = 2$, the relationship is verified.
Verification of Product of Zeroes:
Product of zeroes = $\alpha \cdot \beta = 4 \cdot (-2) = -8$
$\frac{c}{a} = \frac{-8}{1} = -8$
Since $-8 = -8$, the relationship is verified.
Final Answer: The zeroes of the polynomial $x^2 - 2x - 8$ are $4$ and $-2$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $2$ and the product of zeroes is $-8$.
More Questions from Class 10 Mathematics Polynomials EXERCISE 2.2
- 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(iv): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iv) $4u^2 + 8u$
- 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
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