MAIN IDEAS FACTOR POLYNOMIALS. SOLVE POLYNOMIAL EQUATIONS BY FACTORING. 6.6 Solving Polynomial Equations

Types of Factoring 1. GCF 2. Binomial a) Difference of Squares b) Sum of cubes c) Difference of cubes 3. Trinomial 4. Grouping

GCF and Difference of Squares Greatest Common Factor (GCF) Always look for a GCF before trying any other factoring techniques. It will usually make any future factoring easier. Difference of Squaresa² - b² = ( + )( - ) Must be subtraction Even exponent on variables Numbers are perfect squares

Factoring Perfect Cubes Variable has an exponent of 3 All numbers are a perfect cubes Sum of Two cubesa³ + b³ = (a + b)(a² – ab + b²) Difference of Two cubes a³ – b³ = (a – b)(a² + ab + b²) Before using the factoring method you must rewrite the numbers as the appropriate perfect cube. 1) x³ ) y³ – 125 3) y³ + 8x³

Trinomial Factoring Two parentheses ax² + bx + c = ( )( ) Factors of the first term Factors of the last term Combination of factors (O & I) use equal the middle term and determine the signs.

Factoring by Grouping A factoring method that can be used with four term polynomials. Regroup terms 1 & 2 together and 3 & 4 together. Find a GCF for each group If parentheses match you can use factoring by grouping. Write the multiplication problem as ( matching parentheses )( outside parentheses ) 1) x³ + 5x² + 2x ) x² + 3xy + 2xy² + 6y³ 3) y³ – 4y² + 3y – 124) 6a³ – 9a²b - 4ab + 6b²

Factoring Techniques Number of Terms Factoring techniqueGeneral Case Any Number GCF Binomial Difference of Two Squares Sum of Two cubes Difference of Two cubes a² – b²= (a + b)(a – b) a³ + b³ = (a + b)(a² – ab + b²) a³ – b³ = (a – b)(a² + ab + b²) Trinomial General Trinomial ax² + bx + c = (x + p)(x + q) Four or more terms Grouping ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y)

Examples

Methods to solve Polynomial Equations 1. Graphing  Only finds the real solutions. 2. Factoring and Zero Product Property  Factor completely using the factoring techniques we have learned.  Set each factor equal to zero and solve. 3. Quadratic Formula  Can only be used if an equation is in the form ax² + bx + c = 0 *Note the number of solutions (real & imaginary) should match the degree of the polynomial. 1) x⁴ – 29x² = 02) 3x⁴ – 6x³ + 12x² = 0

Examples 3. x³ = 04. x² – 3x – 18 = x³ + 2x² – 4x = 06. x⁴ – 7x² – 8 = 0 7. x³ + 3x² + 4x + 12 = 08. 6x² = 48x