For the Following, Find a Quadratic Polynomial whose Sum and Product Respectively of the Zeros are as Given. Also Find the Zeroes of the Polynomial by Factorization: 21/8, 5/16
By BYJU'S Exam Prep
Updated on: October 17th, 2023
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of the polynomial by factorization:
21/8, 5/16
To find the solution, write the quadratic polynomial in standard form: f(x) = ax^2 + bx + c. Then find the product and sum of the zeroes using the formulas: Product of the zeroes = c/a and Sum of the zeroes = -b/a
Write a quadratic polynomial with the given sum and product of zeroes using the formula:
f(x) = x^2 – (sum of zeroes)x + (product of zeroes)
Thereafter, factorize the quadratic polynomial obtained. This can be done by using any of the following methods:
- a. Trial and error method
- b. Completing the square method
- c. Quadratic formula method
- d. Grouping method
Once you have the quadratic polynomial in factored form, set each factor equal to zero and solve for x to find the zeroes of the polynomial.
Table of content
Solution
We know that the sum of the roots (α + β) and the product of the roots (αβ) are related to the coefficients of the quadratic polynomial through the following formulas:
α + β = -b/a
αβ = c/a
where a, b, and c are the coefficients of the quadratic polynomial ax2 + bx + c.
In this case, we are given the sum of the roots (α + β) and the product of the roots (αβ) as 21/8 and 5/16 respectively. Let’s use these values to find the quadratic polynomial.
Sum of the roots (α + β) = 21/8, so we have:
α + β = -b/a = 21/8
Product of the roots (αβ) = 5/16, so we have:
αβ = c/a = 5/16
We will use the formula x2 – (sum of the zeroes) x + (product of the zeroes) to find the quadratic polynomial whose sum and product of the zeroes are given.
Using this formula, we have:
x2 – (sum of the zeroes) x + (product of the zeroes) = 0
x2 – (21/8)x + (5/16) = 0
we get:
16x2 – 42x + 5 = 0
On solving, we get:
16x2 -(2x + 42x) + 5 = 0
= (2x – 5)(8x – 1) = 0
Therefore, the roots of the quadratic polynomial are:
- α = 5/2
- β = 1/8
So the quadratic polynomial is:(16x2 – 42x + 5)
And its roots are:
x = 5/2 and x = 1/8
Answer
Therefore, the quadratic polynomial whose sum and product of the roots are 21/8 and 5/16 respectively is (16x2 – 42x + 5), and its roots are 5/2 and 1/8.
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