Exercise 25 Page 69

We want to completely factor the given expression. To do so, we will first identify and factor out the greatest common factor (GCF), the common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is 3p.

12p^3-3p

SplitIntoFactors

Split into factors

3p(4p^2)- 3p(1)

FactorOut

Factor out 3p

3p(4p^2-1)

Notice that 1^2=1. Therefore, we can rewrite the given expression as the difference of squares. 3p(4p^2-1^2) ⇔ 3p(2p+1)(2p-1)

Checking Our Answer

Check Our Answer ✓

We can expand our answer and compare it with the given expression.

3p (2p +1) (2p - 1)

Distr

Distribute 3p

(6p^2 +3p) (2p - 1)

Distr

Distribute 6p^2 +3p

2p(6p^2 +3p) - (6p^2 +3p)

Distr

Distribute 2p

12p^3 + 6p^2 - (6p^2 + 3p)

Distr

Distribute -1

12p^3 + 6p^2 - 6p^2 -3p

SubTerm

Subtract term

12p^3 -3p

We can see above that after expanding and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!