Chapter SH, Problem 5.15EP

To determine

Find the remaining factor of the given polynomial.

Answer to Problem 5.15EP

The remaining factors are $\left(x-3\right)$ and $\left(x+2\right)$ .

Explanation of Solution

Given:

The given polynomial is $x^3-2x^2-5x+6$ and the one of its factor is $\left(x-1\right)$ .

Calculation:

Write original expression.

  $x^3-2x^2-5x+6$

Add and subtract $x^2$ in the expression.

  $\begin{array}{l}=x^3-2x^2-x^2+x^2-5x+6\\ =x^3-3x^2+x^2-5x+6\end{array}$

Find the factor. Use factorization method.

  $\begin{array}{l}=x^2\left(x-3\right)+\left(x^2-5x+6\right)\\ =x^2\left(x-3\right)+\left(x^2-3x-2x+6\right)\\ =x^2\left(x-3\right)+\left[x\left(x-3\right)-2\left(x-3\right)\right]\\ =x^2\left(x-3\right)+\left(x-2\right)\left(x-3\right)\\ =\left(x-3\right)\left[x^2+\left(x-2\right)\right]\\ =\left(x-3\right)\left[x^2+x+2\right]\end{array}$

Use factorization method.

  $\begin{array}{l}=\left(x-3\right)\left[x^2+2x-x-2\right]\\ =\left(x-3\right)\left[x\left(x+2\right)-\left(x+2\right)\right]\\ =\left(x-3\right)\left[\left(x+2\right)\left(x-1\right)\right]\\ =\left(x-3\right)\left(x-1\right)\left(x+2\right)\end{array}$

Hence the remaining factors are $\left(x-3\right)$ and $\left(x+2\right)$ .