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

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.

Answer:

If α+β=24 and α-β=8, then the Polynomial having Alpha and Beta as its Zeros is P(x) = x^2 – 24x + 128

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