# 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 Mohit Uniyal|Updated : May 13th, 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
• 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.

## 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 GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com