Chapter 8.7, Problem 35PPS

To determine

To factor: The given polynomial, if possible. If the polynomial cannot be factored, write prime.

Answer to Problem 35PPS

The factored polynomial is $x^3+2x^{2}y-4x-8y=\left(x+2y\right)\left(x-2\right)\left(x+2\right)$ .

Explanation of Solution

Given information:

The given polynomial is $x^3+2x^{2}y-4x-8y$ .

Concept used:

The polynomial of the form $a{c}^3+b{x}^2+cx+d$ are factored by grouping the terms having one or more common factor(s).

Calculations:

The given polynomial E (say) is $E=x^3+2x^{2}y-4x-8y$ that can be written as

  $E=\left(x^3+2x^{2}y\right)-\left(4x+8y\right)$

  $\Rightarrow E=x^2\left(x+2y\right)-4\left(x+2y\right)$    Factor $\left(4m+9\right)$ is common.

  $\Rightarrow E=\left(x+2y\right)\left(x^2-4\right)$

  $\Rightarrow E=\left(x+2y\right)\left({(x)}^2-{(2)}^2\right)$

  $\Rightarrow E=\left(x+2y\right)\left(x-2\right)\left(x+2\right)$    Difference of two squares

Thus, the factored polynomial is $x^3+2x^{2}y-4x-8y=\left(x+2y\right)\left(x-2\right)\left(x+2\right)$ .

The given polynomial is not a prime polynomial.

Conclusion:

The factored polynomial is $x^3+2x^{2}y-4x-8y=\left(x+2y\right)\left(x-2\right)\left(x+2\right)$ .