Questions: Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3 -4 and 1+4i are zeros; f(-1)=60 f(x)= (Type an expression using x as the variable. Simplify your answer.)

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.

n=3

-4 and 1+4i are zeros;

f(-1)=60

f(x)=

(Type an expression using x as the variable. Simplify your answer.)

Transcript text: Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \[ n=3 \] -4 and $1+4 i$ are zeros; \[ \begin{array}{l} f(-1)=60 \\ f(x)=\square \end{array} \] $\square$ (Type an expression using $x$ as the valiable. Simplify your answer.)

Solution

Solution Steps

To find an nth-degree polynomial function with real coefficients, we need to consider the given zeros and the function value. Since complex zeros occur in conjugate pairs, if $1+4i$ is a zero, then $1-4i$ must also be a zero. Thus, the zeros of the polynomial are $-4$, $1+4i$, and $1-4i$. The polynomial can be expressed as $f(x) = a(x + 4)(x - (1+4i))(x - (1-4i))$. We will expand this expression and use the condition $f(-1) = 60$ to solve for the constant $a$.

Step 1: Identify the Zeros

Given the zeros of the polynomial are $-4$, $1 + 4i$, and $1 - 4i$. Since the polynomial has real coefficients, the complex zeros must occur in conjugate pairs.

Step 2: Construct the Polynomial

The polynomial can be expressed as: \[ f(x) = a(x + 4)(x - (1 + 4i))(x - (1 - 4i)) \] Expanding the factors involving the complex zeros: \[ (x - (1 + 4i))(x - (1 - 4i)) = (x - 1 - 4i)(x - 1 + 4i) = (x - 1)^2 + 16 = x^2 - 2x + 17 \] Thus, the polynomial becomes: \[ f(x) = a(x + 4)(x^2 - 2x + 17) \]

Step 3: Expand the Polynomial

Expanding $f(x)$: \[ f(x) = a[(x + 4)(x^2 - 2x + 17)] = a(x^3 + 2x^2 + 9x + 68) \]

Step 4: Use the Given Condition

We know that $f(-1) = 60$. Substituting $-1$ into the polynomial: \[ f(-1) = a[(-1)^3 + 2(-1)^2 + 9(-1) + 68] = a[-1 + 2 - 9 + 68] = a[60] \] Setting this equal to 60 gives: \[ 60a = 60 \implies a = 1 \]

Step 5: Final Polynomial

Substituting $a = 1$ back into the polynomial: \[ f(x) = x^3 + 2x^2 + 9x + 68 \]