Unit 4: Factoring and Complex Numbers Warm-Up: Complete the half-sheet on finding GCF. How can you use the calculator to find factors of a number? Use the functions “y=“ or “gcd”

When factoring: always find GCF first if it exists Examples: 8x – 4 14x3 – 7x2 6x2 + 12x – 21 4n4 + 6n3 – 8n2 9 – 27x2

X-box Factoring

X- Box ax2 + bx + c Trinomial (Quadratic Equation) Product of a & c ax2 + bx + c Fill the 2 empty sides with 2 numbers that are factors of ‘a·c’ and add to give you ‘b’. b

X- Box x2 + 9x + 20 Trinomial (Quadratic Equation) Fill the 2 empty sides with 2 numbers that are factors of ‘a·c’ and add to give you ‘b’. 5 4 9

X- Box 2x2 -x - 21 Trinomial (Quadratic Equation) -42 Fill the 2 empty sides with 2 numbers that are factors of ‘a·c’ and add to give you ‘b’. 6 -7 -1

X-box Factoring This is a guaranteed method for factoring quadratic equations—no guessing necessary! We will learn how to factor quadratic equations using the x-box method

LET’S TRY IT! Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Objective: I can use the x-box method to factor non-prime trinomials.

First and Last Coefficients Factor the x-box way y = ax2 + bx + c GCF GCF Product ac=mn First and Last Coefficients 1st Term Factor n* GCF n m Middle Last term Factor m* b=m+n Sum GCF Factors “n” and “m” come from the “X” so include “x” in the box with them.

Factor the x-box way x2 -5x -10 2x Example: Factor x2 -3x -10 (1)(-10)= -10 x -5 x2 -5x x GCF -5 2 -3 2x -10 +2 GCF GCF GCF x2 -3x -10 = (x-5)(x+2)

Factor the x-box way Example: Factor 3x2 -13x -10 x -5 -30 3x 3x2 -15x +2 3x2 -13x -10 = (x-5)(3x+2)

Examples Factor using the x-box method. 1. x2 + 4x – 12 x +6 x2 6x x a) b) x +6 -12 4 x2 6x -2x -12 x 6 -2 -2 Solution: x2 + 4x – 12 = (x + 6)(x - 2)

Examples continued 2. x2 - 9x + 20 x -4 x x2 -4x -5x 20 a) b) x -4 -4 -5 20 -9 x x2 -4x -5x 20 -5 Solution: x2 - 9x + 20 = (x - 4)(x - 5)

Examples continued 3. 2x2 - 5x - 7 2x -7 x 2x2 -7x 2x -7 a) b) 2x -7 -14 -5 -7 2 x 2x2 -7x 2x -7 +1 Solution: 2x2 - 5x – 7 = (2x - 7)(x + 1)

Examples continued 3. 15x2 + 7x - 2 3x +2 5x 15x2 10x -3x -2 a) b) 3x +2 -30 7 5x 15x2 10x -3x -2 10 -3 -1 Solution: 15x2 + 7x – 2 = (3x + 2)(5x - 1)

Extra Practice x2 +4x -32 4x2 +4x -3 3x2 + 11x – 20

Notes What do you do if there are no factors that add to equal “b” and multiply to give “ac”?