construct the polynomial function

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Construct a polynomial function with the following properties: third degree , only real coefficients, -3 and 2 + I are two of the zeros, y-intercepts is -30. Please explain how to get the answer and what the answer is.

May 6th, 2015

Since all coefficients of the polynomial are real and one of the zeros is 2 + i, the polynomial must also have a conjugate zero 2 - i. The degree of the polynomial is 3, so it must have three zeros and can be written as 

P(x) = A(x - x_1)(x - x_2)(x - x_3) where A is a constant.

Write P(x) = A(x - (-3))(x - 2 - i)(x - 2 + i) = 

A(x + 3)(x - 2 - i)(x - 2 + i) =           use the formula (a - b)(a + b) = a^2 - b^2 where a = x - 2, b = i

A (x + 3)[(x - 2)^2 - i^2] =                 use i^2 = -1

A(x + 3) (x^2 - 4x + 4 + 1) = A(x + 3)(x^2 - 4x + 5)

To find the A, use the y-intercept P(0) = -30: P(0) = 3*5*A = - 30, and A = -2.

Thus, the polynomial is P(x) = -2(x + 3)(x^2 - 4x + 5) = -2(x^3 - 4x^2 + 5x + 3x^2 - 12x + 15) =

-2(x^3 - x^2 - 7x + 15) = -2x^3 + 2x^2 + 14x - 30 

May 6th, 2015

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