Chapter SH, Problem 5.16EP

To determine

Find the remaining factor of the given polynomial.

Answer to Problem 5.16EP

The remaining factors are $\left(x-4\right)$ and $\left(2x+1\right)$ .

Explanation of Solution

Given:

The given polynomial is $2x^3-x^2-25x-12$ and the one of its factor is $\left(x+3\right)$ .

Calculation:

Write original expression.

  $2x^3-x^2-25x-12$

Add and subtract $6x^2$ in the expression.

  $\begin{array}{l}=2x^3+6x^2-6x^2-x^2-25x-12\\ =2x^3+6x^2-7x^2-25x-12\end{array}$

Find the factor. Use factorization method.

  $\begin{array}{l}=2x^2\left(x+3\right)-\left(7x^2+25x+12\right)\\ =2x^2\left(x+3\right)-\left(7x^2+21x+4x+12\right)\\ =2x^2\left(x+3\right)-\left[7x\left(x+3\right)+4\left(x+3\right)\right]\\ =2x^2\left(x+3\right)-\left(x+3\right)\left(7x+4\right)\\ =\left(x+3\right)\left[2x^2-\left(7x+4\right)\right]\\ =\left(x+3\right)\left[2x^2-7x-4\right]\end{array}$

Use factorization method.

  $\begin{array}{l}=\left(x+3\right)\left[2x^2-8x+x-4\right]\\ =\left(x+3\right)\left[2x\left(x-4\right)+\left(x-4\right)\right]\\ =\left(x+3\right)\left[\left(2x+1\right)\left(x-4\right)\right]\\ =\left(x-4\right)\left(2x+1\right)\left(x+3\right)\end{array}$

Hence the remaining factors are $\left(x-4\right)$ and $\left(2x+1\right)$ .