Solving Polynomials Review of Previous Methods Factoring Cubic Expressions U-Substitution

Basic Steps  Set the polynomial equal to zero  Factor  Set each factor equal to zero  Solve each factor  CHECK THE ANSWERS!  Substitute the answer into the original equation & verify you get the same value on both sides of the equal sign 2

Greatest Common Factor (GCF)  Look for the largest common number and/or variables in each term  Divide each term by the common terms Factor: 3

Rainbow Method (2 nd degree Polynomial) 4

Grouping  Solve as you would the last 3 steps of the “Rainbow” Method  Only works with 4 terms that GCF does not apply to  Steps:  Write polynomial in standard form  Use GCF on the first 2 terms  Use GCF on the last 2 terms Note: You MUST have repeated expressions in parenthesis  Otherwise, it’s prime or was factored incorrectly  Rewrite the factors outside the parenthesis inside their own parenthesis  Write the repeated expression once in parenthesis 5

Grouping Example  Factor 6

Special Cases 7 Factoring FormulasExample Difference of two Squares:9a 2 – 16 =(3a) 2 – (4) 2 = (3a + 4)(3a – 4) Difference of two Cubes: x 3 – y 3 = (x - y)(x 2 + xy + y 2 ) 8a 3 – 27 = (2a) 3 – (3) 3 = (2a – 3)[(2a) 2 + (2a)(3) + (3) 2 ] =(2a – 3)(4a 2 + 6a + 9) Sum of two Cubes: x 3 + y 3 = (x + y)(x 2 – xy + y 2 ) 125a = (5a) 3 + (1) 3 = (5a + 1)[(5a) 2 – (5a)(1) + (1) 2 ]

Difference of Squares 8  Must have a “-”

Difference of Cubes  Must have a “-” 9

Sum of Cubes  Must have a “+” 10

U-Substitution  Typically used when other methods are close to working, but not quite right  Used in calculus…a lot  Steps:  Replace the variable with the letter “u” to a power that would make another method work  Factor  Re-substitute the original value for “u”  Factor again if possible & solve 11

U-Substitution Example  Factor 12