Chapter 8.7, Problem 37PPS

To determine

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

Answer to Problem 37PPS

The factored polynomial is $2r^3-r^2-72r+36=\left(2r-1\right)\left(r-6\right)\left(r+6\right)$ .

Explanation of Solution

Given information:

The given polynomial is $2r^3-r^2-72r+36$ .

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 $E=2r^3-r^2-72r+36$ that can be written as

  $E=\left(2r^3-r^2\right)-\left(72r-36\right)$

  $\Rightarrow E=r^2\left(2r-1\right)-36\left(2r-1\right)$

  $\Rightarrow E=\left(2r-1\right)\left(r^2-36\right)$       Factor $\left(b^2-1\right)$ is common.

  $\Rightarrow E=\left(2r-1\right)\left({(r)}^2-{(6)}^2\right)$

  $\Rightarrow E=\left(2r-1\right)\left(r-6\right)\left(r+6\right)$       Difference of two squares

Thus, the factored polynomial is $2r^3-r^2-72r+36=\left(2r-1\right)\left(r-6\right)\left(r+6\right)$ .

The given polynomial is not a prime polynomial.

Conclusion:

The factored polynomial is $2r^3-r^2-72r+36=\left(2r-1\right)\left(r-6\right)\left(r+6\right)$ .