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

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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