Chapter 4.1, Problem 46AYU

In Problems 43-50, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

Zeros: $-3$, $-1$, 2, 5; degree 4.

To determine

To form: A polynomial function whose real zeros and degree are given.

Answer to Problem 46AYU

$f\left(x\right)=\left( x+\text{ 3}\right)\left(x+1\right)\left(x-2\right)\left(x-\text{ 5}\right)=x^4-3x^3-15x^2+19x+\text{ 30}$

Explanation of Solution

Given:

Zeros $=-3,-1,2,5$.

Degree $=\text{ 4}$.

If $r$ is the real zero of a polynomial function then $\left(x-r\right)$ is a factor of $f$.

Zeros $=-3,-1,2,5$.

Degree $=\text{ 4}$.

The polynomial function $=f\left(x\right)=a \left(x+\text{ 3}\right)\left(x+1\right)\left(x-2\right)\left(x-\text{ 5}\right)$.

Here $a\text{ =}$ leading coefficient and is a non zero real number.

Value of causes a stretch, compression or reflection but it does not affect the $x\text{-intercepts}$.

If $a\text{ = 1}$,

$f\left(x\right)=1\left(x+3\right)\left(x+1\right)\left(x-2\right)\left(x-5\right)=x^4-3x^3-15x^2+19x+30$

The $x\text{-intercepts }=-3,-1,2,5$.