Chapter 8.6, Problem 62PFA

a.

To determine

To factor: The polynomial $z^2+2yz-3y^2$

Answer to Problem 62PFA

  $\left(z+3y\right)\left(z-y\right)$

Explanation of Solution

Given:

  $z^2+2yz-3y^2$

Calculation:

The given expression is $z^2+2yz-3y^2$

Write the middle term $2yz=3yz-yz$

  $z^2+3yz-yz-3y^2$

Make groups as shown below

  $\left(z^2+3yz\right)+\left(-yz-3y^2\right)$

Factored out the GCF from each group

  $z\left(z+3y\right)-y\left(z+3y\right)$

Factored out the common term

  $\left(z+3y\right)\left(z-y\right)$

b.

To determine

To factor: The polynomial $3z^2+2yz-y^2$

Answer to Problem 62PFA

  $\left(z+y\right)\left(3z-y\right)$

Explanation of Solution

Given:

  $3z^2+2yz-y^2$

Calculation:

The given expression is $3z^2+2yz-y^2$

Write the middle term $2yz=3yz-yz$

  $3z^2+3yz-yz-y^2$

Make groups as shown below

  $\left(3z^2+3yz\right)+\left(-yz-y^2\right)$

Factored out the GCF from each group

  $3z\left(z+y\right)-y\left(z+y\right)$

Factored out the common term

  $\left(z+y\right)\left(3z-y\right)$

c.

To determine

To factor: The polynomial $27y^2-12z^2$

Answer to Problem 62PFA

  $3\left(3y+2z\right)\left(3y-2z\right)$

Explanation of Solution

Given:

  $27y^2-12z^2$

Calculation:

The given expression is $27y^2-12z^2$

Factored out the GCF

  $3\left(9y^2-4z^2\right)$

Write in perfect squares form

  $3\left({\left(3y\right)}^2-{\left(2z\right)}^2\right)$

Apply the difference of squares formula $a^2-b^2=\left(a+b\right)\left(a-b\right)$

  $3\left(3y+2z\right)\left(3y-2z\right)$