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Q2(i):
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) $\frac{1}{4}, -1$
Solution :
Given:
The sum of the zeroes of the quadratic polynomial ($\alpha + \beta$) = $\frac{1}{4}$
The product of the zeroes of the quadratic polynomial ($\alpha \cdot \beta$) = $-1$
To Find:
A quadratic polynomial $p(x)$ that satisfies the given conditions.
Step 1: Understanding the Relationship between Zeroes and Coefficients
A quadratic polynomial in the variable $x$ is generally expressed in the form $p(x) = ax^2 + bx + c$, where $a \neq 0$.
According to the relationship between the zeroes and coefficients of a quadratic polynomial:
Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Step 2: Formulating the Polynomial
Any quadratic polynomial can be expressed in terms of its sum and product of zeroes using the following identity:
$p(x) = k[x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})]$
where $k$ is a non-zero real constant.
Step 3: Substituting the Given Values
Substitute the given values into the identity:
$p(x) = k[x^2 - (\frac{1}{4})x + (-1)]$
$p(x) = k[x^2 - \frac{1}{4}x - 1]$
Step 4: Simplifying the Expression
To obtain a polynomial with integer coefficients, we can choose a suitable value for the constant $k$. Let $k = 4$ (the denominator of the fraction):
$p(x) = 4[x^2 - \frac{1}{4}x - 1]$
Distribute the $4$ across the terms inside the bracket:
$p(x) = 4(x^2) - 4(\frac{1}{4}x) - 4(1)$
$p(x) = 4x^2 - 1x - 4$
$p(x) = 4x^2 - x - 4$
Step 5: Verification (Optional but Recommended)
For $p(x) = 4x^2 - x - 4$:
Sum of zeroes = $-\frac{b}{a} = -(\frac{-1}{4}) = \frac{1}{4}$ [Matches the given sum]
Product of zeroes = $\frac{c}{a} = \frac{-4}{4} = -1$ [Matches the given product]
Final Answer: The required quadratic polynomial is $4x^2 - x - 4$.
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(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(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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