# If α+β=24 and α-β=8. Find the Polynomial having Alpha and Beta as its Zeros. Also Verify the Relationship between the Zeros and the Coefficients of the Polynomial

By BYJU'S Exam Prep

Updated on: October 17th, 2023

Polynomial is P(x) = x^2 – 24x + 128 that is having Alpha and Beta as its Zeros.

Here are the steps used to find the polynomial and verify the relationship between the zeroes and the coefficients:
We will first add the two equations to eliminate the β term and then simplify the equation for α.
Then we will subtract the second equation from the first to eliminate the α term and and then simplify the equation for β.
Then we will substitute values of α and β into the polynomial expression P(x) = (x – α)(x – β) top get the desired polynomial.
To verify the relationship between the zeroes and the coefficients, we will use the following relationships: Sum of the zeroes and Product of the zeroes

## If α+β=24 and α-β=8. Find the Polynomial having Alpha and Beta as its Zeros. Also Verify the Relationship between the Zeros and the Coefficients of the Polynomial

Solution:

To find the polynomial with α and β as its zeros, we can use the fact that if α and β are the zeros of a quadratic polynomial, then the polynomial can be expressed as:

P(x) = (x – α)(x – β)

In this case, we are given that α + β = 24 and α – β = 8. We can use these equations to find the values of α and β and then substitute them into the polynomial expression. Let’s solve for α and β:

Adding the two equations: (α + β) + (α – β) = 24 + 8 2α = 32 α = 16

Subtracting the two equations: (α + β) – (α – β) = 24 – 8 2β = 16 β = 8

Now, we have the values of α and β. Let’s substitute them into the polynomial expression:

P(x) = (x – α)(x – β) P(x) = (x – 16)(x – 8)

Expanding the brackets:

P(x) = x^2 – 8x – 16x + 128 P(x) = x^2 – 24x + 128

Therefore, the polynomial with α and β as its zeros is P(x) = x^2 – 24x + 128.

Verification:

Now, let’s verify the relationship between the zeroes (α and β) and the coefficients of the polynomial.

From the quadratic polynomial P(x) = x^2 – 24x + 128, we can observe the following relationships:

Sum of the zeroes (α + β) = -(-24) = 24 (which matches the given value) Product of the zeroes (α×β) = 16×8 = 128 (which matches the constant term in the polynomial)

Thus, we can see that the relationship between the zeroes (α and β) and the coefficients of the polynomial holds true.