Chapter 4.6, Problem 48E

To determine

To write: and graph a polynomial function of degree 5 that has all positive or negative real zeros, and to write the function in standard form.

Answer to Problem 48E

The polynomial in standard form is $f\left(x\right)=x^5-3x^4-5x^3+15x^2+4x+12$ .

Explanation of Solution

Given information:

Label each $x$ intercept.

Calculation:

The polynomial function of degree 5 that have all positive or negative zero is,

  $f\left(x\right)=\left(x-1\right)\left(x+1\right)\left(x-2\right)\left(x+2\right)\left(x-3\right)$

The graph of function is shown below:

  

Now write the polynomial in standard form.

  $\begin{array}{c}f\left(x\right)=\left(x^2-1\right)\left(x^2-4\right)\left(x-3\right)\\ =\left(x^4-5x^2+4\right)\left(x-3\right)\\ =x\left(x^4-5x^2+4\right)-3\left(x^4-5x^2+4\right)\\ =x^5-3x^4-5x^3+15x^2+4x+12\end{array}$

Thus, the polynomial in standard form is $f\left(x\right)=x^5-3x^4-5x^3+15x^2+4x+12$ .