Chapter 8.7, Problem 28PPS

To determine

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

Answer to Problem 28PPS

The factored polynomial is $4m^3+9m^2-36m-81=\left(4m+9\right)\left(m-3\right)\left(m+3\right)$ .

Explanation of Solution

Given information:

The given polynomial is $4m^3+9m^2-36m-81$ .

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 $E=4m^3+9m^2-36m-81$ .

  $E=\left(4m^3+9m^2\right)-\left(36m+81\right)$

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

  $\Rightarrow E=\left(4m+9\right)\left(m^2-9\right)$

  $\Rightarrow E=\left(4m+9\right)\left({(m)}^2-{(3)}^2\right)$

  $\Rightarrow E=\left(4m+9\right)\left(m-3\right)\left(m+3\right)$       Difference of two squares

Thus, the factored polynomial is $4m^3+9m^2-36m-81=\left(4m+9\right)\left(m-3\right)\left(m+3\right)$ .

Conclusion:

The factored polynomial is $4m^3+9m^2-36m-81=\left(4m+9\right)\left(m-3\right)\left(m+3\right)$ .