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Q2(v):
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(v) $-\frac{1}{4}, \frac{1}{4}$
Solution :
Given: The sum of the zeroes ($\alpha + \beta$) is $-\frac{1}{4}$ and the product of the zeroes ($\alpha \cdot \beta$) is $\frac{1}{4}$.
To Find: A quadratic polynomial $p(x)$ that satisfies the given conditions.
Step 1: Understanding the Relationship between Zeroes and Coefficients
For any quadratic polynomial of the form $ax^2 + bx + c$, where $a \neq 0$, the relationship between the zeroes ($\alpha, \beta$) and the coefficients is given by the following formulas:
Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$
Product of zeroes: $\alpha \cdot \beta = \frac{c}{a}$
Step 2: Formulating the General Quadratic Polynomial
A quadratic polynomial can be expressed in terms of the sum and product of its 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 $\alpha + \beta = -\frac{1}{4}$ and $\alpha \cdot \beta = \frac{1}{4}$ into the identity:
$p(x) = k\left[x^2 - \left(-\frac{1}{4}\right)x + \left(\frac{1}{4}\right)\right]$
[Simplifying the signs: $-(- \frac{1}{4}) = +\frac{1}{4}$]
$p(x) = k\left[x^2 + \frac{1}{4}x + \frac{1}{4}\right]$
Step 4: Simplifying the Expression
To obtain a polynomial with integer coefficients, we can choose $k = 4$ (or any multiple of the denominator):
$p(x) = 4\left[x^2 + \frac{1}{4}x + \frac{1}{4}\right]$
Distributing the $4$ across the terms inside the brackets:
$p(x) = 4(x^2) + 4\left(\frac{1}{4}x\right) + 4\left(\frac{1}{4}\right)$
$p(x) = 4x^2 + x + 1$
Final Answer: The required quadratic polynomial is $4x^2 + x + 1$.
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(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(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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