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Q2(vi):
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(vi) $4, 1$
Solution :
Given:
The sum of the zeroes of the quadratic polynomial ($\alpha + \beta$) = $4$.
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
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 standard 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 any non-zero real constant.
Step 3: Substituting the Given Values
Substitute the given values into the identity:
Sum of zeroes = $4$
Product of zeroes = $1$
$p(x) = k[x^2 - (4)x + (1)]$
Step 4: Simplifying the Expression
By choosing the simplest case where $k = 1$, we obtain the polynomial:
$p(x) = x^2 - 4x + 1$
Justification:
If we verify the zeroes of $p(x) = x^2 - 4x + 1$:
Sum of zeroes = $-\frac{b}{a} = -\frac{-4}{1} = 4$ [Matches the given sum]
Product of zeroes = $\frac{c}{a} = \frac{1}{1} = 1$ [Matches the given product]
Final Answer: The required quadratic polynomial is $x^2 - 4x + 1$.
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(v): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (v) $-\frac{1}{4}, \frac{1}{4}$
CBSE Solutions for Class 10 Mathematics Polynomials
Chapters in CBSE - Class 10 Mathematics
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