Questions: Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Fourth-degree and a single zero of -1.

$Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Fourth-degree and a single zero of -1.$

Transcript text: Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms. Fourth-degree and a single zero of -1.

Solution

Solution Steps

To construct a fourth-degree polynomial with a single zero at $ x = -1 $, we can use the fact that if $ x = -1 $ is a zero, then $ (x + 1) $ is a factor of the polynomial. Since the polynomial is fourth-degree, we can express it as $ (x + 1)^4 $. This ensures that the polynomial has a single zero at $ x = -1 $ with multiplicity four.

Step 1: Define the Polynomial

To construct a fourth-degree polynomial with a single zero at $ x = -1 $, we start with the factor $ (x + 1) $. Since it is a fourth-degree polynomial, we express it as: \[ P(x) = (x + 1)^4 \]

Step 2: Expand the Polynomial

Next, we expand the polynomial $ P(x) $: \[ P(x) = (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1 \]