Questions: Find all the zeros (real and complex) of P(x)=x^4+3x^3+18x^2+48x+32 Separate answers with commas. Use exact values, including fractions and radicals, instead of decimals. Enter complex numbers in the form a+bi. Zeros: Write P in factored form as a product of linear and irreducible quadratic factors. Do not use complex linear factors. Be sure to write the full equation, including "P(x)=". Function:

$Find all the zeros (real and complex) of P(x)=x^4+3x^3+18x^2+48x+32 Separate answers with commas. Use exact values, including fractions and radicals, instead of decimals. Enter complex numbers in the form a+bi. Zeros: Write P in factored form as a product of linear and irreducible quadratic factors. Do not use complex linear factors. Be sure to write the full equation, including "P(x)=". Function:$

Transcript text: Find all the zeros (real and complex) of \[ P(x)=x^{4}+3 x^{3}+18 x^{2}+48 x+32 \] Separate answers with commas. Use exact values, including fractions and radicals, instead of decimals. Enter complex numbers in the form $a+b i$. Zeros: $\square$ Write $P$ in factored form as a product of linear and irreducible quadratic factors. Do not use complex linear factors. Be sure to write the full equation, including " $P(x)=$ ". Function: $\square$

Solution

Solution Steps

To find the zeros of the polynomial $ P(x) = x^4 + 3x^3 + 18x^2 + 48x + 32 $, we can use numerical methods or symbolic computation to find both real and complex roots. Once the roots are found, we can express the polynomial in its factored form using these roots. The factored form will include linear factors for real roots and irreducible quadratic factors for pairs of complex conjugate roots.

Step 1: Identify the Zeros of the Polynomial

The polynomial $ P(x) = x^4 + 3x^3 + 18x^2 + 48x + 32 $ has the following zeros: \[ -2, -1, -4i, 4i \] These include two real zeros, $-2$ and $-1$, and two complex zeros, $-4i$ and $4i$.

Step 2: Write the Polynomial in Factored Form

Using the zeros identified, the polynomial can be expressed in its factored form. The real zeros contribute linear factors, and the complex conjugate zeros contribute an irreducible quadratic factor: \[ P(x) = (x + 1)(x + 2)(x^2 + 16) \]