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REVIEW:. Seven Steps for Factoring a Quadratic Polynomial (or a polynomial in quadratic form). Step #1. Factor out a: G reatest C ommon F actor ( This is sort of like the distributive property in reverse.)
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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.
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 aBinomial 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
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”.
Step #7 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? Identifya,b, and cin: 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