Chapter 4.4, Problem 12E

To determine

To factor: the polynomial completely.

Answer to Problem 12E

The factorized form is $18v^9+33v^8+14v^7=v^7\left(3v+2\right)\left(6v+7\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $18v^9+33v^8+14v^7$

Calculation:

Consider the given polynomial.

  $18v^9+33v^8+14v^7$

Take out $v^7$ common for each term.

  $18v^9+33v^8+14v^7=v^7\left(18v^2+33v+14\right)$

Now, break the coefficient of middle term $33$ as sum or difference of two numbers such that product of the numbers is equal to the product of first term and constant term that is product of $18$ and $14$ .

Such numbers will be $21$ and $12$ .

Now, factorize $18v^2+33v+14$ as shown:

  $\begin{array}{c}18v^2+33v+14=18v^2+21v+12v+14\\ =3v\left(6v+7\right)+2\left(6v+7\right)\\ =\left(3v+2\right)\left(6v+7\right)\end{array}$

Therefore, $18v^9+33v^8+14v^7=v^7\left(3v+2\right)\left(6v+7\right)$