1 Exponent Rules and Monomials Standards 3 and 4 Simplifying Monomials: Problems POLYNOMIALS Monomials and Polynomials: Adding and Multiplying Multiplying Binomials: FOIL with MODELING Multiplying Polynomials with MODELING Dividing Polynomials: Long Division Synthetic Division of Polynomials Greatest Common Factor: GCF Factoring Polynomials: 2 Terms with MODELING Factoring Polynomials: Perfect Square Trinomials with MODELING Factoring Polynomials: General END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 3: Students are adept at operations on polynomials, including long division. STANDARD 4: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 ESTÁNDAR 3: Los estudiantes son capaces de hacer operaciones de polinomios, incluyendo division larga. ESTÁNDAR 4: Los estudiantes factorizan diferencias de cuadrados, trinomios cuadrados perfectos, y la suma y diferencia de dos cubos. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 Standards 3 and 4 MONOMIALS Negative Exponents: a = -n n 1 a = a -n n 1 a and For any real number a, and any integer n, where a = a 6 1 x a = -2 x = -6 y = y = z -3 = b z 7 1 b PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 Standards 3 and 4 MONOMIALS a m a n = a m+n Multiplying Powers: For any real number a and integers m and n = x 3+5 = x 8 x 3 5 y y y 2 47 = y = y 13 a m a n = a m-n Dividing Powers: For any real number a, except a=0, and integers m and n = x 8-3 = x 5 =y 9-8 x x 8 3 y y 9 8 = y PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 Standards 3 and 4 MONOMIALS a m n = a mn Power of a Power: Suppose m and n are integers and a and b are real numbers. Then the following is true: = x (4) (3) = x 12 x 4 3 y 5 7 = y (5) (7) = y 35 Power of a Product: (ab) n = a b nn = x y 55 (xy) 5 (-3pr) 3 = (-3) p r = -27p r 3 3 Power of a Quotient: a b n = a b n n a b -n = b a n n b a n = = y x (2)(3) (3)(3) = y x 6 9 y x y x x y = = x y (2)(5) (3)(5) = x y Power to the zero: a 0 = 1 (4y) 0 (-3kp) 0 = 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Standards 3 and 4 (4x y )(-2x y z ) = (4)(-2)x x y y z = -8x y z 3+4 = -8x y z p r w p r w x Finding the GCF between 18 and 36: = = We take all the numbers that repeat with the least exponent: GCF= = 18 p r w x = = p r w x p 2w x = -18p r w p r w x (4x y )(-2x y z ) Simplify the following monomials: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 Standards 3 and 4 (3k n )(-7k n r ) = (3)(-7)k k n n r 5 6 = -21k n r = -21k n r a b c a b c d Finding the GCF between 27 and 48: = 3 3 We take all the numbers that repeat with the least exponent: 3 GCF= a b c d = = a b c d a b c a b c d (3k n )(-7k n r ) = a 16bd = c 9 Simplify the following monomials: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 a b c a b c d = 2 b c d (-3)(-2) (-5)(-2) (-1)(-2) (-3)(-2) = b c d = b c d Simplify the following monomial: = a b c d a b c d = STANDARD 10 a b c a b c d PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 Standards 3 and 4 It is possible to add or subtract terms of a polynomial only if they are LIKE TERMS: Simplify 5xy + 6z x -9xy + 10z x – 15z xy + 6z x -9xy + 10z x – 15z Simplify -8a b c + 7b c - 3b c + a b c a b c + 7b c - 3b c + a b c = -8a b c + a b c + 7b c – 3b c = -7a b c + 7b c -3b c It is possible to use the distributive property of multiplication over addition to multiply polynomials: Simplify 4x(2x y + 3x y – 6x y ) = (4x)(2x y) + (4x)(3x y ) + (4x)(-6x y ) x(2x y + 3x y – 6x y ) = (4)(2)x y + (4)(3)x y + (4)(-6)x y =8x y + 12x y -24x y = -7a b c - 3b c +7b c = 5xy – 9xy + 6z x + 10z x -15z = -4xy +16z x – 15z 5 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 (2x +1)(x + 4) (4) x (1) 2x x +2x +1 (4) + F O I L = 2x + 9x = 2x + 8x + x = Standards 3 and 4 Simplify the following expressions: (6x +3)(2x + 5) (5) (2x) (3) 6x (2x) +6x +3 (5) + F O I L =12x + 36x =12x + 30x + 6x = (6x - 3)(x + 5) (5) x (-3) 6x x +6x +(-3)(5) + F O I L = 6x + 27x = 6x + 30x -3x = (4x - 3)(3x - 7) (-7) (3x) (-3) 4x (3x) +4x +(-3) (-7) + F O I L =12x - 37x =12x -28x - 9x = First Outer Inner Last: FOIL Method. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 Area of a Rectangle L A = L W where: W= width L= length A= area W PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 (2x +1)(x + 4) (4) x (1) 2x x +2x +1 (4) + F O I L = 2x + 9x = 2x + 8x + x = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x x + 4 2x + 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 Simplify the following expressions: (2x - 2)(3x - 1) (-1) (3x) (-2) 2x (3x) +2x+(-2) (-1) + F O I L = 6x - 8x = 6x -2x - 6x = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x 3x – 1 2x – 2 x x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 (2x – 2)(x + 3) (3) x (-2) 2x x +2x +(-2) (3) + F O I L = 2x + 4x – 6 2 = 2x + 6x -2x – 6 2 = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 (2x – 2)(x + 3) (3) x (-2) 2x x +2x +(-2) (3) + F O I L = 2x + 4x – 6 2 = 2x + 6x -2x – 6 2 = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 (2x – 2)(x + 3) (3) x (-2) 2x x +2x +(-2) (3) + F O I L = 2x + 4x – 6 2 = 2x + 6x -2x – 6 2 = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 Standards 3 and 4 x+1x+1 X +5-3x +5x -3x 2 x 2 x x x 3 -2x 2 x- 4x x-4x-4 X x +7x -4x 2 2 x x x 3 -8x 2 x- 3x Simplify x+1 Simplify x- 4x x-4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 Standards 8, 10, 11 L L L B V = Bh B = (L)(L) B= L 2 V = L L 2 V= L 3 VOLUME OF A CUBE: REVIEW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 STANDARDS x1x1 = 1 3 = 1 1 CUBED 2x2x2 = 2 3 = 8 2 CUBED 3x3x3 = 3 3 = 27 3 CUBED 4x4x4 = 4 3 =64 4 CUBED What is the volume for these cubes? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 x x x x x x V = (x) V = (x) (1) V = (x) (1) V = (1) = x 3 2 = 1 Lets find the volume for this prisms: Can we use this knowledge to multiply polynomials? PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 STANDARD Multiply: (x+3)(x+2)(x+1) (x+2) (x+1) (x+3) (x + 2)(x + 3) (3) x (2) x x + x + (2) (3) + F O I L = x + 5x = x + 3x +2x = x+1x+1 X +6+5x +6x +5x 2 x 2 x x x 3 +6x 2 x +5x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 STANDARD Multiply: (x+3)(x+2)(x+1) = x + 6x + 11x (x + 2)(x + 3) (3) x (2) x x + x + (2) (3) + F O I L = x + 5x = x + 3x +2x = x+1x+1 X +6+5x +6x +5x 2 x 2 x x x 3 +6x 2 x +5x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24 STANDARD Multiply: (x+3)(x+2)(x+1) = x + 6x + 11x (x + 2)(x + 3) (3) x (2) x x + x + (2) (3) + F O I L = x + 5x = x + 3x +2x = x+1x+1 X +6+5x +6x +5x 2 x 2 x x x 3 +6x 2 x +5x (x+2) (x+1) (x+3) So, a third degree polynomial may be represented GEOMETRICALLY, by the VOLUME OF A RECTANGULAR PRISM, in this case with SIDES (x+3), (x+2) and (x+1). PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25 STANDARD Multiply: (2x+1)(x+3)(x+4) (x+3) (x+4) (2x+1) (2x + 1)(x + 3) (3) x (1) 2x x +2x + (1) (3) + F O I L = 2x + 7x = 2x + 6x +1x = x+4x+4 X x + 3x +7x 2 8x 2 2x x 2x 3 +15x 2 2x +7x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26 STANDARD Multiply: (2x+1)(x+3)(x+4) (2x + 1)(x + 3) (3) x (1) 2x x +2x + (1) (3) + F O I L = 2x + 7x = 2x + 6x +1x = x+4x+4 X x + 3x +7x 2 8x 2 x 3 2x +7x x 2x 3 +15x x 2x 3 +15x 2 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

27 STANDARD Multiply: (2x+1)(x+3)(x+4) (2x + 1)(x + 3) (3) x (1) 2x x +2x + (1) (3) + F O I L = 2x + 7x = 2x + 6x +1x = x+4x+4 X x + 3x +7x 2 8x 2 x 3 2x +7x x 2x 3 +15x x 2x 3 +15x 2 = (x+3) (x+4) (2x+1) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

28 Standards 3 and 4 x - 4x x - x -7x x x 3 - 4x 2 + 5x - 3x 2 -12x x 2 -12x x -10x x -12x +46x x x 3 -10x 2 +26x - -2x 2 +20x x 2 +20x Divide by x - x -7x x - 4x Divide by x -10x x -12x +46x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

29 Standards 3 and Divide x + 2x -20x + 24 by x-2 using synthetic division 2 3 with x- (+2) x + 4x Divide x +x - 8x + 16 by x+4 using synthetic division 2 3 with x- (-4) x - 3x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

30 Standards 3 and 4 Factoring the Greatest Common Factor (GCF): 4x y z - 16x y z + 32x = 4xx y z - 4(4x)xy z + 8(4x) =4x(x y z – 4xy z + 8) p q r + 9p q r - 3pqr = (3)(-9)pp qq r + (3)(3)ppqqrr -3pqrr =3pqr(-9 p q + 3pqr – r ) 2 22 = (3pqr)(-9)p q + (3pqr)(3qpr)-(3pqr)r 2 22 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

31 Standards 3 and 4 Difference of Two Squares: (x+2)(x-2) x - 4= 2 a - b = (a+b)(a-b) 2 2 9y - 64= 2 (3y+8)(3y-8) Sum of Two Cubes: a + b = (a+b)(a -ab + b ) y + 27z = 33 64k +125j = 33 Difference of Two Cubes: a - b = (a-b)(a +ab + b ) y - z = 33 27k - j = 3 3 (2y + 3z)((2y) - (2y)(3z) + (3z) ) 22 (2y + 3z)(4y - 6yz + 9z ) 2 2 = (4k + 5j)((4k) - (4k)(5j) + (5j) ) 22 (4k + 5j)(16k - 20kj + 25j ) 2 2 = (6y - z)((6y) + (6y)(z) + (z) ) 22 (3k - j)(9k + 3kj + j ) 2 2 = (3k - j)((3k) + (3k)(j) + (j) ) 22 (6y - z)(36y + 6yz + z ) 2 2 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

32 Difference of Two Squares: (x+2)(x-2) STANDARD SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

33 Difference of Two Squares: (x+2)(x-2) STANDARD SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

34 Difference of Two Squares: (x+2)(x-2) = x STANDARD SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

35 STANDARD x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

36 STANDARD x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

37 STANDARD x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

38 STANDARD x x x + 3 x – 3 (x+3)(x-3) = x – 9 2 SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

39 Standards 3 and 4 Perfect Square Trinomials: a + 2ab + b = (a + b) a - 2ab + b = (a - b) x + 4x = (x +2) 2 x + 6x = (x +3) 2 = (x) + 2(x)(2) + (2) 2 2 = (x) + 2(x)(3) + (3) x + 40x = (5x) + 2(5x)(4) + (4) 2 2 = (5x + 4) 2 x -10x = (x - 5) 2 x - 14x = (x -7) 2 = (x) - 2(x)(5) + (5) 2 2 = (x) - 2(x)(7) + (7) x - 64x = (8x) - 2(8x)(4) + (4) 2 2 = (8x - 4) 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

40 STANDARD SPECIAL PRODUCTS (x +2) 2 = (x) + 2(x)(2) + (2) 2 2 x + 4x = x x x +2 1 = (x+2)(x+2) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

41 STANDARD SPECIAL PRODUCTS (x +3) 2 = (x) + 2(x)(3) + (3) 2 2 x + 6x = x x x +3 1 = (x+3)(x+3) 1 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

42 Standards 3 and 4 General Trinomials: B -5B (B+5) (B-10) Two numbers that multiplied be negative fifty should be (+)(-) or (-)(+) Two numbers that added be negative 5 should be |(-)| >| (+)| (1)(-50) 1+(-50)= -49 (5)(-10) 5+(-10)= -5 (2)(-25) 2+(-25)= x Factor the following trinomial: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

43 Standards 3 and x (4)(-15) = x - 11x x + (4-15)x x + 4x -15x x(3x)+ (4x)1 -5(3x) + (-5)(1) 4x(3x+1) – 5 (3x +1) (4x- 5)(3x+1) Factor the following trinomial: Find two numbers that multiplied be (12)(-5)=-60 and added -11. (3)(-20) = -17 (2)(-30) = -28 (1)(-60) = x - 11x -5 2 General Trinomials: acx + (ad + bc)x + bd = (ax +b)(cx +d) 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

44 Standards 3 and x -6x +11x x + (3+8)x x + 3x +8x x(2x)- (-3x)1 +4(2x) + (4)(-1) -3x(2x-1) + 4(2x -1) (-3x+ 4)(2x-1) Factor the following trinomial: Find two numbers that multiplied be (-6)(-4)= +24 and added -11. (3)(8) 3 + 8= 11 (2)(12) = 14 (1)(24) = 25 -6x +11x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

45 Standards 3 and x (8)(-12) = -4 8x - 4x x + (8-12)x x + 8x -12x x(4x)+ (2x)4 -3(4x) + (-3)(4) 2x(4x+4) – 3 (4x +4) (2x- 3)(4x+4) Factor the following trinomial: Find two numbers that multiplied be (8)(-12)=-96 and added -4. 8x - 4x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved