Factor and Solve Polynomial Equations Homework Questions? Chapter 5.4 Factor and Solve Polynomial Equations Homework Questions?

Warm up! Multiply the polynomials 𝑥+2 𝑥+3 (2𝑥−1)(2𝑥+1) 𝑥+2 𝑥+3 (2𝑥−1)(2𝑥+1) (𝑥−7 ) 2 3 𝑥 2 (𝑥+5)

Special Case Polynomials Factoring Polynomials Difference of Squares 𝒂 𝟐 − 𝒃 𝟐 =(𝒂−𝒃)(𝒂+𝒃) 𝟗 𝒙 𝟐 −𝟏= 𝟑𝒙−𝟏 𝟑𝒙+𝟏 Sum and Difference of Cubes 𝒂 𝟑 + 𝒃 𝟑 =(𝒂+𝒃)( 𝒂 𝟐 −𝒂𝒃+ 𝒃 𝟐 ) 𝟖 𝒙 𝟑 +𝟐𝟕=(𝟐𝒙+𝟑)(𝟒 𝒙 𝟐 −𝟔𝒙+𝟗) 𝒂 𝟑 − 𝒃 𝟑 =(𝒂−𝒃)( 𝒂 𝟐 +𝒂𝒃+ 𝒃 𝟐 ) 𝟔𝟒 𝒙 𝟑 −𝟏=(𝟒𝒙−𝟏)(𝟏𝟔 𝒙 𝟐 −𝟒𝒙+𝟏) Binomial Square (Perfect Square Trinomials) 𝒂 𝟐 +𝟐𝒂𝒃+ 𝒃 𝟐 =(𝒂+𝒃 ) 𝟐 𝒙 𝟐 +𝟔𝒙+𝟗=(𝒙+𝟑 ) 𝟐 𝒂 𝟐 −𝟐𝒂𝒃+ 𝒃 𝟐 =(𝒂−𝒃 ) 𝟐 𝟒 𝒙 𝟐 −𝟏𝟐𝒙+𝟗=(𝟐𝒙−𝟑 ) 𝟐 Binomial Cubes (Perfect Cube Polynomials) 𝒂 𝟑 +𝟑 𝒂 𝟐 𝒃+ 𝟑𝒂𝒃 𝟐 + 𝒃 𝟑 =(𝒂+𝒃 ) 𝟑 𝒙 𝟑 +𝟔 𝒙 𝟐 +𝟏𝟐𝒙+𝟖=(𝒙+𝟐 ) 𝟑 𝒂 𝟑 −𝟑 𝒂 𝟐 𝒃+ 𝟑𝒂𝒃 𝟐 − 𝒃 𝟑 =(𝒂−𝒃 ) 𝟑 𝒙 𝟑 −𝟗 𝒙 𝟐 +𝟐𝟕𝒙−𝟐𝟕=(𝒙−𝟑 ) 𝟑 Factoring (Grouping) 𝒙 𝟐 +𝟓𝒙+𝟒=(𝒙+𝟒)(𝒙+𝟏) 𝟐𝒙 𝟐 −𝟑𝒙−𝟐𝟎=(𝟐𝒙+𝟓)(𝒙−𝟒) Common Factor (Factor Out) 𝒙 𝟐 +𝒙=𝒙(𝒙+1) 𝟖𝒙 𝟐 +𝟐𝟎𝒙=𝟒𝒙(𝟐𝒙+𝟓)

Sum and Difference of Cubes Factoring with Sum and Difference of Cubes Sum and Difference of Cubes 𝒂 𝟑 + 𝒃 𝟑 =(𝒂+𝒃)( 𝒂 𝟐 −𝒂𝒃+ 𝒃 𝟐 ) 𝟖 𝒙 𝟑 +𝟐𝟕=(𝟐𝒙+𝟑)(𝟒 𝒙 𝟐 −𝟔𝒙+𝟗) 𝒂 𝟑 − 𝒃 𝟑 =(𝒂−𝒃)( 𝒂 𝟐 +𝒂𝒃+ 𝒃 𝟐 ) 𝟔𝟒 𝒙 𝟑 −𝟏=(𝟒𝒙−𝟏)(𝟏𝟔 𝒙 𝟐 −𝟒𝒙+𝟏) 𝑥 3 +64= 8𝑥 3 +27 𝑥 3 −64= 16𝑥 5 −250 𝑥 2

X box (i.e. products of c, sums of b) Factoring Flow Chart Is it a polynomial? Can you factor out a GCF? Type of Polynomial? Factor out! Monomial Fin! Put it standard from Fin! Binomial Trinomial Terms >3 Sum Difference Squares Not Factorable Difference of squares X box (i.e. products of c, sums of b) Not Factorable Factor a=1 Factor by Grouping Not Factorable Factor Sum Difference Cubes Sum of Cubes Difference of a≠1 Factor by Grouping Not Factorable Factor Always Check Your Solution by Multiplying the Solution (i.e. FOIL)

Factoring by Grouping when # terms >3 𝑥 3 − 3𝑥 2 −16𝑥+48 27𝑥 3 +45 𝑥 2 −3𝑥−5 (𝑥 3 − 3𝑥 2 )+(−16𝑥+48) (27𝑥 3 +45 𝑥 2 )+(−3𝑥−5) 𝑥 2 (𝑥−3)−16(𝑥−3) 9𝑥 2 (3𝑥+5)−1(3𝑥+5) (𝑥 2 −16)(𝑥−3) (9𝑥 2 −1)(3𝑥+5) (𝑥−4)(𝑥+4)(𝑥−3) (3𝑥−1)(3𝑥+1)(3𝑥+5)

Factoring the following completely 2𝑥 3 − 𝑥 2 +2𝑥−1 4𝑥 3 +16 𝑥 2 −𝑥−4 𝑥 3 +64 4𝑥 6 −20 𝑥 4 +24 𝑥 2 𝑥 3 +7 𝑥 2 −9𝑥−63 16 𝑥 4 −625