Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

3, -13, and 5 + 4i



@ganeshie8

Answer 1

@Shay17

Answer 2

A polynomial with real coefficients and known roots \(x_1,x_2,\cdots,x_n\) will have the form
\[a(x-x_1)(x-x_2)\cdots(x-x_n)\]
where \(a\) is a real number. Because all coefficients are real, any complex roots will occur in conjugate pairs.

Since there's one complex root \(5+4i\), that must mean that the polynomial also has \(5-4i\) as a root.

The remaining roots are real, so the polynomial will look like
\[a(x-3)(x+13)(x-5-4i)(x-5+4i)\]
"Standard form" seems to mean "expand" so that it looks like
\[a_0+a_1x+a_2x^2+\cdots+a_nx^n\]