CPM Homework Help : INT3 Problem 8-120

Spud has done it again. He has lost another polynomial function. This one is a cubic, written in standard form. He knows that there were two complex zeros, $–2 ± 5i,$ and one real zero, $–1$ . What could his original function have been?

Step 1:

Use the three zeros to write the polynomial in factored form.

$p\left(x\right) = \left(x − \left(− 1\right)\right)\left(x − \left(−2 + 5i\right)\right)\left(x − \left(−2 − 5i\right)\right)$

Step 2:

Multiply the two complex polynomials.

$\left(x − \left(−2 + 5i\right)\right)\left(x − \left(−2 − 5i\right)\right)$
$x^{2} + 4x + 29$

Step 3:

Multiply the result by $\left(x + 1\right)$ .

$\left(x + 1\right)\left(x^{2} + 4x + 29\right)$

Answer:

$p\left(x\right) = x^{3} + 5x^{2} + 33x + 29$