Find a polynomial function of lowest degree with integer coefficients, whose leading coefficient is one, and that has the zeros 8, , and 3 i .

Question

Find a polynomial function of lowest degree with integer coefficients, whose leading coefficient is one, and that has the zeros 8,  , and 3 i .

Quizplus · Accepted Answer

For a polynomial function with integer coefficients and roots 8, 11ebfdc1_ec29_cd74_abe3_ddcb7e759728_TBX8727_11, and 3i, we also need to include its complex conjugate, which is -3i. Therefore, the factors of this polynomial function are (x - 8), (x - 11ebfdc1_ec29_cd74_abe3_ddcb7e759728_TBX8727_11), (x - 3i), and (x + 3i). Multiplying these factors, we get:
(x - 8)(x - 11ebfdc1_ec29_cd74_abe3_ddcb7e759728_TBX8727_11)(x - 3i)(x + 3i)
Expanding this expression will result in a polynomial function of degree 4 with integer coefficients. However, we can simplify it by noticing that the factors (x - 3i) and (x + 3i) multiply to (x^2 + 9). Therefore, the polynomial function becomes:
(x - 8)(x - 11ebfdc1_ec29_cd74_abe3_ddcb7e759728_TBX8727_11)(x^2 + 9)
Expanding this expression gives us a polynomial function of degree 3 with integer coefficients and leading coefficient 1. The only choice that matches this polynomial function is B.

Find a Polynomial Function of Lowest Degree with Integer Coefficients