Chapter 8.7, Problem 21PPS

To determine

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

Answer to Problem 21PPS

The factored polynomial is $12m^3-22m^2-70m=2m\left(2m-7\right)\left(3m+5\right)$ .

Explanation of Solution

Given information:

The given polynomial is $12m^3-22m^2-70m$ .

Concept used:

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

To factor expression $\mathrm{A }{x}^2+\mathrm{B }x+\mathrm{C }$ , we find two factors of $\mathrm{A }\times \mathrm{C }$ such that their sum equals B .

Calculations:

The given polynomial E (say) is $E=12m^3-22m^2-70m$ that can be written as

  $\Rightarrow E=2m\left(6m^2-11m-35\right)$

Now, to factor $\left(m^2-11m-35\right)$ , we find two factors of $6\times 35$ such that their difference is -11.

  $6\times 35=2\times 3\times 5\times 7$

  $\Rightarrow 6\times -35=-\left(2\times 5\right)\times \left(3\times 7\right)$

  $\Rightarrow 6\times -35=\left(10\right)\times \left(-21\right)$ , such that $-11=-21+10$

  $\Rightarrow E=2m\left(6m^2-21m+10m-35\right)$

  $\Rightarrow E=2m\left(3m(2m-7)+5(2m-7)\right)$

  $\Rightarrow E=2m(2m-7)\left(3m+5\right)$

  $\Rightarrow E=2m\left(2m-7\right)\left(3m+5\right)$

Thus, the factored polynomial is $12m^3-22m^2-70m=2m\left(2m-7\right)\left(3m+5\right)$ .

The given polynomial is not a prime polynomial.

Conclusion:

The factored polynomial is $12m^3-22m^2-70m=2m\left(2m-7\right)\left(3m+5\right)$ .