Exercise 62 Page 222

To completely factor the given expression, we will first identify the greatest common factor.

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 3.

Factor the Trinomial

We want to factor a perfect square trinomial. 3( 4x^2+12x+9) How do we know that the expression is a perfect square trinomial? Well, let's ask a few questions.

Is the first term a perfect square?	4x^2= 2^2x^2=( 2x)^2 ✓
Is the last term a perfect square?	9= 3^2 ✓
Is the middle term twice the product of 3 and 2x?	12x=2* 3* 2x ✓

As we can see, the answer to all three questions is yes! Therefore, we can write the trinomial as the square of a binomial. Note that there is an addition sign in the middle.

3(4x^2+12x+9)

▼

a^2+2ab+b^2=(a+b)^2

WritePow

Write as a power

3(2^2x^2+12x+3^2)

ProdPowII

a^m b^m = (a b)^m

3((2x)^2+12x+3^2)

SplitIntoFactors

Split into factors

3((2x)^2+2(3)(2)x+3^2)

CommutativePropMult

Commutative Property of Multiplication

3((2x)^2+2(2)x(3)+3^2)

AssociativePropMult

Associative Property of Multiplication

3((2x)^2+2(2x)(3)+3^2)

FacPosPerfectSquare

a^2+2ab+b^2=(a+b)^2

3(2x+3)^2

Checking Our Answer

Checking Our Answer ✓

Let's un-factor our answer and compare it with the given expression.

3 (2 x + 3)^2

ExpandPosPerfectSquare

(a+b)^2=a^2+2ab+b^2

3 (4 x^2 + 12 x + 9 )

Distr

Distribute 3

12x^2 + 36 x + 27

After expanding and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!