Section 5.4

Objectives…  Be able to factor “completely” a quadratic expression by using any of the following methods: 1) GCMF (Greatest Common Monomial Factor) 2) “X-Box Method” for factoring quadratic trinomials 3) Formula for a perfect square trinomial 4) Formula for a difference of two squares

What exactly is “factoring”?  “Factoring” is when an expression is rewritten as a product of its factors  GCMF: - to find the GCMF, you must determine the following:  GCF of the coefficients of each term  the lowest exponent with any of the common variables - put them together…you have your GCMF!

GCMF Factoring…

Let’s try some…

Factoring Quadratic Trinomials…

Factoring Quadratic Trinomials by the “X-Box” Method…  Procedure: 1) Determine the product by multiplying the coefficient of the quadratic term with the constant 2) Determine the sum by identifying the coefficient of the linear term 3) Determine the two numbers that when multiplied get you the product and when added get you the sum (some thinking is required here!)

Factoring Quadratic Trinomials by the “X-Box” Method…  Procedure continued… 4) Draw an “X” (product goes in upper wedge, sum goes in lower wedge)- when you find your #’s, they go into side wedges 5) Make a box: - put in the box the following terms: first term, terms with the coefficients found in step #3, last term 6) Find the GCMF of each row and column

Factoring Quadratic Trinomials by the “X-Box” Method…  7) Put both the horizontal and vertical GCMF’s together in parentheses…you have your factorization!

Let’s do some…

Factoring Perfect Square Trinomials

How do I know when a trinomial is a “perfect square”?  Perform a short test on the trinomial: 1) Are both the first and last terms “perfect squares” ? (coefficients are perfect squares and all of the exponents are even) 2) When you multiply the SQUARE ROOTS of the first and last terms together and then DOUBLE the product, do you get the middle term? (Signs are NOT factored into the answer to this question) 3) If the answer to both questions was “yes”, then you have a perfect square trinomial!

How to factor “perfect square trinomials”…

Factoring a Difference of Two Squares

How to factor a “difference of two squares”…

Let’s factor some of these…

Let’s factor the ones that were…

Let’s factor a few “differences of two squares”…

Factoring “Completely”…  factoring “completely” means to factor until ALL of the factors are prime (may have to factor multiple times in order to achieve this)  Procedure for factoring completely: 1) Find the GCMF and factor it out from the initial expression 2) Look at the expression remaining in the parentheses and determine whether it can be factored any further (if you can, factor it using the appropriate method)

Factoring “Completely”…  continue with step #2 until all of the factors are prime (all have a GCMF of 1)

Let’s do some…

How about factoring these two?