If α and β are the Zeros of the Quadratic Polynomial p(x) = 4x^2 − 5x − 1, Find the Value of α^2β + αβ^2

If α and β are the Zeros of the Quadratic Polynomial p(x) = 4x² − 5x − 1, Find the Value of α²β + αβ².

Solution:

To find the value of α²β + αβ², we need to substitute the zeros of the quadratic polynomial p(x) = 4x² - 5x - 1 into the expression.

From Vieta's formulas

We have the sum of the zeros is equal to -b/a

α + β = 5/4 and

the product of the zeros is equal to c/a

αβ = -1/4

Now, we can calculate α²β + αβ²:

α²β + αβ² = αβ(α + β) + αβ(α + β)

Substituting the values we know:

= (-1/4)(5/4) = -5/16

Therefore, the value of α²β + αβ² is -5/16.

Answer:

For Quadratic Polynomial p(x) = 4x² − 5x − 1, the value of α²β + αβ² is -5/16

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