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The Distributive Property (of * over +) a(b+c)=ab+ac Geometric Proof b a ab c a ac Geometric Proof a b c ab ac a Geometric Proof b+c a ab+ac a Geometric Proof b+c a a(b+c) a a(b+c)=ab+ac Combine like terms • • • • 2x+5x 2x+5x (2+5)x 7x ba+ca (b+c)a Don’t combine unlike terms • • • • 2x2+5x 2x2+5x (2x+5)x stuck ba+ca (b+c)a Distribute the negative • • • • • 2-(x-4) 2+-1(x+-4) 2+(-1)x+(-1)(-4) 2-x+4 6-x turn - into +a(b+c) ab+ac The distributive property works for any size sum a(b+c+d+e+…) =ab+ac+ad+ae+a… Simplify completely: (4 x 5x 7) (3x 2 x 9) 2 2 a)12x2-3x-2 b)7x2-7x+16 c) 12x2-7x+16 d)7x2-3x+16 e)None of the above (4x - 5x + 7) - (-3x + 2x - 9) 2 2 (4x - 5x + 7) + -1(-3x + 2x - 9)Distribute the negative 2 2 (4x 2 - 5x + 7) + (3x 2 - 2x + 9) 4x - 5x + 7 + 3x - 2x + 9 2 2 Combine like terms 7x - 7x +16 2 B What most of you call FOILing • • • • • • • • • • (x-6)(x+2) (x-6)(x+2) a(b+c) (x-6)x+(x-6)2 ab+ac (x+-6)x+(x-6)2 (b+c)a xx+-6x +(x-6)2 ba+ca x2-6x +(x+-6)2 (b+c)a x2-6x +2x+(2)(-6) ab+ac x2+-6x +2x-12 ba+ca x2+(-6+2)x-12 (b+c)a x2+-4x-12 Geometric “FOILing” x b x a x2 ax bx ab (x+a)(x+b)=x2+ax+bx+ab Lattice Method (2x3-3x2+2x+1)(-4x2+6)=? 2x3 -3x2 +2x +1 -4x2 -8x5 +12x4 -8x3 -4x2 +6 12x3 -18x2 +12x +6 =-8x5+12x4+4x3-22x2+12x+6 n n Find the product: (a + 6)(a - 6) 2n a) a - 6 2 n b) a - 6 2 n c) a - 36 2 n d) a + 36 e) None of the above n n Find the product: (a + 6)(a - 6) Remember rules of exponents! anan=an+n=a2n an +6 an a2n 6an -6 -6an -36 2n a - 36 E) None of the above Factor and Factoring • A factor is something being multiplied • 2 and x are factors of 2x • Factoring is the art of turning something into a multiplication. • Ex: 6 can be factored into 2*3: • 2 and 3 are factors of 6. • Factoring nicely can be hard because you have to add information. Factoring Trinomials pt1 • (x+a)(x+b) • x2+(a+b)x+ab • x2-5x+6 = (x+a)(x+b) • what are a and b? Factoring Trinomials pt1 • (x+a)(x+b) • x2+(a+b)x+ab • • • • • x2+-5x+6 = (x+a)(x+b) what are a and b? a+b=-5 ab=6 a=-2, b=-3 (x-2)(x-3) Factoring Trinomials pt2 • • • • (cx+a)(dx+b) cdx2+(da+cb)x+ab 2x2+7x-15 = (cx+a)(dx+b) what are a, b, c and d? Factoring Trinomials pt2 • • • • • • • • • • (mx+a)(nx+b) mnx2+(na+mb)x+ab 2x2+7x-15 = (mx+a)(nx+b) what are a, b, m and n? mn=2 na+mb=7 ab=-15 m=2, n=1, a=-5, b=3? (1)(-5)+2(3)=1 m=2, n=1, a=5, b=-3? (1)(5)+2(-3)=-1 m=2, n=1, a=3, b=-5? (1)(3)+2(-5)=-7 m=2, n=1, a=-3, b=5? (1)(-3)+2(5)=7 (2x-3)(x+5) Factoring By Grouping • • • • • 3x3-6x2+9x-18 (3x3-6x2)+(9x-18) 3x2(x-2)+9(x-2) (3x2+9)(x-2) The trick is to pull something out of each group so that what is left over is the same. Factoring By Grouping • The trick is to pull something out of each group so that what is left over is the same. • 4x3-6x2-10x+15 • 4x3-6x2+-10x+15 • (4x3-6x2)+(-10x+15) • 2x2(2x-3)+5(-2x+3) Not the same! • 2x2(2x-3)+-5(2x-3) The same! • (2x2-5)(2x+3) Factor: 3 x 7 x 20 2 a) b) c) d) e) (x-5)(x+4) (3x+5)(x-4) (3x-5)(x+4) (x-4)(3x+5) None of the above 3 x 7 x 20 2 3x2+7x-20 (mx+a)(nx+b)= mnx2+(na+mb)x+ab mn=3 m=3, n=1 ab=-20 a=2, b=-10; a=-2, b=10; a=-10, b=2; a=10, b=-2; a=4, b=-5; a=-4, b=5 a=-5, b=4; a=5, b=-4 … 1a+3b=7 1(-5)+3(4)=-5+12=7 a=-5, b=4 c) (3x-5)(x+4)