REVIEW: Seven Steps for Factoring a Quadratic Polynomial (or a polynomial in quadratic form)

Step #1 Factor out a: Greatest Common Factor (This is sort of like the distributive property in reverse.) Start with the numerical coefficients and the constant and look for a common (number) factor in all of them. Then look for a common variable in all terms and factor out the lowest value exponent of that variable. Factor out both the number and letter common factors by using steps similar to division. The remaining quotients from each term stay behind in the parentheses containing the group.

(matching) (leftovers) Step #2 If the polynomial is a “4 – termer”: FACTOR BY GROUPING Make two binomial groups JOINED by ADDITION Factor out GCF from each group (individually) Factor out GCF from ENTIRE expression Final answer should be…. (matching) (leftovers)

What if you have a Binomial or Trinomial ? Place the polynomial in standard form for a quadratic expression: ax² + bx + c Where a is the coefficient of the quadratic term (x²), b is the coefficient of the linear term (x), and c is the constant term, signs included.

Step #3 Identify a , b , and c .

Step #4 Multiply a · c

Find a pair of factors of ac Step #5 Find a pair of factors of ac that combine to equal b . [ Combine can mean to add or subtract depending on the signs of the factors ]

Step #6 BUST “b” Rewrite the polynomial, busting the middle term into two terms that add to it. (use the numbers you just found instep #5 as your coefficients) This will force the original polynomial to now be a “4 – termer”.

( This is the same as in step #2 ) FACTOR BY GROUPING ( This is the same as in step #2 )

Seven Steps for Factoring a Quadratic (or quadratic form) Polynomial… Factor out GCF “4 – termer” ?, Factor By Grouping Trinomial? Identify a , b , and c in: ax² + bx + c Multiply a · c Find a pair of factors of ac that combine to equal b Rewrite as a “4 – termer” Factor By Grouping

Help for Step #5 If ac is positive; both factors will have the same sign as b If ac is negative; only the biggest factor will have the same sign as b , (and the smaller factor will have the opposite sign)

Factor: x² + 6x +8 Step #1…. No GCF Step #2…. Not a “4 – termer” Step #3…. a = 1 , b = 6 , c = 8 Step #4…. ac = 8 Step #5…. Factors of 8; 1•8 & 2•4 ..that combine to equal b (6); 2 & 4 Step #6…. Rewrite: x² + 2x + 4x + 8

Factor: x² + 6x +8 cont. Step #7…. Factor by Grouping: x² + 2x + 4x + 8 ( x² + 2x ) + ( 4x + 8 ) x ( x + 2 ) + 4 ( x + 2 ) ( x + 2 )( x + 4 )

Seven Steps… Factor out GCF “4 – termer” ?, Factor By Grouping Identify a , b , and c in: ax² + bx + c Multiply a · c Find a pair of factors of ac that combine to equal b If ac is positive; both factors will have the same sign as b If ac is negative; only the biggest factor will have the same sign as b , (and the smaller factor will have the opposite sign) Rewrite as a “4 – termer” Factor By Grouping