Chapter 6.7, Problem 14CYU

To determine

To Find: The polynomial function of the least degree with integral coefficients that have the given zeroes.

Answer to Problem 14CYU

The polynomial function is $P\left(x\right)=x^4-5x^3+5x^2+5x-6$ .

Explanation of Solution

Given:

The given zeros are $3,-1,1,2$ .

Calculation:

From the given zeroes, consider the given polynomial function is,

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

Then, use the foil method to expand the above equation as,

  $\begin{array}{c}P\left(x\right)=x\left(x\right)+x\left(1\right)+\left(-3\right)\left(x\right)+\left(-3\right)\left(1\right)\\ =x^2+x-3x-3\\ =x^2-2x-3\end{array}$

Now multiply the third binomial with the result as,

  $\begin{array}{c}P\left(x\right)=\left(x^2-2x-3\right)\left(x-1\right)\\ =x\left(x^2-2x-3\right)+\left(-1\right)\left(x^2-2x-3\right)\\ =x^3-2x^2-x^2-3x+2x+3\\ =x^3-3x^2-x+3\end{array}$

Now multiply the fourth binomial with the result as,

  $\begin{array}{c}P\left(x\right)=\left(x^3-3x^2-x+3\right)\left(x-2\right)\\ =x\left(x^3-3x^2-x+3\right)-2\left(x^3-3x^2-x+3\right)\\ =x^4-3x^3-2x^3-x^2+6x^2+3x+2x-6\\ =x^4-5x^3+5x^2+5x-6\end{array}$

Thus, the polynomial function is $P\left(x\right)=x^4-5x^3+5x^2+5x-6$ .