Factoring Polynomials 133
Step 4. Factor the common factor 3x out of the fi rst term and the common
factor 2 out of the second term.
= 3 x(((()− )− 2 ()−
Step 5. Use the distributive property to factor (3x − 2) from the expression.
= ( 3 x − 2 )( 3 x − 2 )
Step 6. Write the factored form.
91 x^21224 x 4 =()(( 333322
Perfect Trinomial Squares
The trinomial 9x^2 − 12 x + 4, which equals (3x − 2)^2 , is a perfect trinomial
square. If you recognize that ax^2 +bx+c is a perfect trinomial square,
then you can factor it rather quickly. A trinomial is a perfect square if a and
c are both positive and bac. The following problem illustrates the
procedure.
Problem Factor.
a. 4 x^2 − 20 x + 25
b. x^2 + 6 x + 9
Solution
a. 4 x^2 − 20 x + 25
Step 1. Identify the coeffi cients a, b, and c and check whether bac.
a = 4, b = −20, and c = 25
|b| = |−20| = 20 and 22 ac 2 ⋅ 422525 ⋅ 252 = 20
Thus, 4x^2 − 20 x + 25 is a perfect trinomial square.
Step 2. Indicate that 4x^2 − 20 x+ 25 will factor as the square of a binomial.
4 x^2 − 20 x+ 25 = ( )^2
Step 3. Fill in the binomial. The fi rst term is the square root of 4x^2 and the
last term is the square root of 25. The sign in the middle is the same
as the sign of the middle term of the trinomial.
42 xx^2222002005 =()−^2 is the factored form.
