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Q1(v):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(v) $t^2 - 15$
Solution :
Given: A quadratic polynomial $p(t) = t^2 - 15$.
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 $p(t)$, we set $p(t) = 0$.
$t^2 - 15 = 0$
We can rewrite this equation using the algebraic identity $a^2 - b^2 = (a - b)(a + b)$.
Since $15 = (\sqrt{15})^2$, we have:
$t^2 - (\sqrt{15})^2 = 0$
$(t - \sqrt{15})(t + \sqrt{15}) = 0$
[By the Zero Product Property, if $ab = 0$, then $a=0$ or $b=0$]
Therefore, $t - \sqrt{15} = 0$ or $t + \sqrt{15} = 0$.
$t = \sqrt{15}$ or $t = -\sqrt{15}$.
The zeroes of the polynomial are $\alpha = \sqrt{15}$ and $\beta = -\sqrt{15}$.
Step 2: Identifying coefficients
Comparing the given polynomial $t^2 - 15$ with the standard quadratic form $at^2 + bt + c$:
$a = 1$
$b = 0$ (since there is no $t$ term)
$c = -15$
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 = $\alpha + \beta = \sqrt{15} + (-\sqrt{15}) = 0$
Coefficient ratio = $-\frac{b}{a} = -\frac{0}{1} = 0$
Since $0 = 0$, the relationship is verified.
Verification of Product of Zeroes:
Product of zeroes = $\alpha \cdot \beta = (\sqrt{15}) \cdot (-\sqrt{15}) = -(\sqrt{15})^2 = -15$
Coefficient ratio = $\frac{c}{a} = \frac{-15}{1} = -15$
Since $-15 = -15$, the relationship is verified.
Final Answer: The zeroes of the polynomial $t^2 - 15$ are $\sqrt{15}$ and $-\sqrt{15}$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $0$ and the product of zeroes is $-15$.
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(iv): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iv) $4u^2 + 8u$
- 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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