Exercise 42 Page 61

To factor the given expression we will start by identifying and factoring out the greatest common factor (GCF). Then we will use factoring by grouping and the formula for factoring the difference of perfect squares.

Factor Out the GCF

The greatest common factor (GCF) of an expression is a 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 x.

Factoring by Grouping

Now, to factor by grouping, we will take the greatest common factor of the first pair of terms and the greatest common factor of the second pair of terms.

Notice that (x+6) is a factor of both terms, so we can factor it out.

Difference of Squares

Now, look closely at the expression x^2-36. It can be expressed as the difference of two perfect squares.

Recall the formula to factor a difference of squares. a^2- b^2 ⇔ ( a+ b)( a- b) We can apply this formula to our expression. x( x^2- 6^2 )(x+6) ⇕ x( x+ 6)( x- 6)(x+6) Since factor (x+6) appears in the factored expression twice, we can write it as x(x+6)^2(x-6).