Exercise 24 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 w^2.

w^4-w^2

SplitIntoFactors

Split into factors

w^2(w^2)- w^2(1)

FactorOut

Factor out w^2

w^2(w^2-1)

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

Checking Our Answer

Check Our Answer ✓

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

w^2 (w +1) (w - 1)

Distr

Distribute w^2

(w^3 +w^2) (w - 1)

Distr

Distribute w^3 +w^2

w(w^3 +w^2) - (w^3 +w^2)

Distr

Distribute w

w^4 + w^3 - (w^3 + w^2)

Distr

Distribute -1

w^4 + w^3 - w^3 -w^2

SubTerm

Subtract term

w^4 -w^2

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!