1. 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. 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. 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. 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 = 0 2 1 a 6 1 x a = -2 x = -6 y = -8 8 1 y = z -3 = b -7 3 1 z 7 1 b PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5. 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 2+4+7 = 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. 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 3 3 3 = -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 2 3 3 y x 3 2 -5 x y 2 3 5 = = x y (2)(5) (3)(5) = x y 10 15 Power to the zero: a 0 = 1 (4y) 0 (-3kp) 0 = 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7. 7 Standards 3 and 4 (4x y )(-2x y z ) 5 3 4 2 2 2 3 4 = (4)(-2)x x y y z 5 2 2+5 2 = -8x y z 3+4 = -8x y z 7 7 2 -18p r w 3 42 36p r w x 5 3 4 2 18 2 9 3 33 1 2 9 3 33 1 36 2 Finding the GCF between 18 and 36: 2 2 3 2 3 2 18 = 3 2 2 36 = 2 2 3 2 We take all the numbers that repeat with the least exponent: 2 3 2 GCF= = 18 p r w x 4-2 3-3 2-4 -5 = -18.. 18 36.. 18 = p r w x 2 -5 0 -22 1 p 2w x 5 2 2 - = -18p r w 3 42 36p r w x 5 3 4 2 (4x y )(-2x y z ) 5 3 4 2 2 Simplify the following monomials: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8. 8 Standards 3 and 4 (3k n )(-7k n r ) 5 2 2 6 7 7 2 2 = (3)(-7)k k n n r 5 6 = -21k n r 6+5 7 2+2 = -21k n r 11 4 7 -27a b c 5 79 48a b c d 2 6 8 3 27 3 9 3 33 1 24 2 12 6 3 48 2 Finding the GCF between 27 and 48: 3 3 27 = 3 3 We take all the numbers that repeat with the least exponent: 3 GCF= a b c d 7-3 5-6 9-8 -2 = -27.. 3 48.. 3 = a b c d -9 16 -2 14 -27a b c 5 79 48a b c d 2 6 8 3 (3k n )(-7k n r ) 5 2 2 6 7 1 2 2 2 4 3 48 = 2 4 3 a 16bd 2 4 - = c 9 Simplify the following monomials: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9. 9 a b c 3 24 2 a b c d 5 6 7 2 -2 1 = 2 b c d (-3)(-2) (-5)(-2) (-1)(-2) (-3)(-2) = b c d 66 10 4 = b c d 66 10 2 2 Simplify the following monomial: = a b c d -5 -3 0 2 -2 a b c d 2-2 3-6 4-7 -5 -2 2 = STANDARD 10 a b c 3 24 2 a b c d 5 6 7 2 -2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10. 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 5 5 3 5xy + 6z x -9xy + 10z x – 15z 5 5 3 Simplify -8a b c + 7b c - 3b c + a b c 3 5 3 5 3 5 2 5 -8a b c + 7b c - 3b c + a b c 3 5 3 5 3 5 2 5 = -8a b c + a b c + 7b c – 3b c 3 5 3 5 3 5 2 5 = -7a b c + 7b c -3b c 3 5 3 5 2 5 It is possible to use the distributive property of multiplication over addition to multiply polynomials: Simplify 4x(2x y + 3x y – 6x y ) 5 43 3 2 = (4x)(2x y) + (4x)(3x y ) + (4x)(-6x y ) 5 4 2 3 3 4x(2x y + 3x y – 6x y ) 5 43 3 2 = (4)(2)x y + (4)(3)x y + (4)(-6)x y 1+5 1+4 1+3 2 3 =8x y + 12x y -24x y 6 5 2 4 3 = -7a b c - 3b c +7b c 3 5 2 5 3 5 = 5xy – 9xy + 6z x + 10z x -15z 5 5 3 = -4xy +16z x – 15z 5 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11. 11 (2x +1)(x + 4) (4) x (1) 2x x +2x +1 (4) + F O I L = 2x + 9x + 4 2 = 2x + 8x + x + 4 2 = 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 + 15 2 =12x + 30x + 6x + 15 2 = (6x - 3)(x + 5) (5) x (-3) 6x x +6x +(-3)(5) + F O I L = 6x + 27x - 15 2 = 6x + 30x -3x - 15 2 = (4x - 3)(3x - 7) (-7) (3x) (-3) 4x (3x) +4x +(-3) (-7) + F O I L =12x - 37x + 21 2 =12x -28x - 9x + 21 2 = First Outer Inner Last: FOIL Method. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12. 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. 13 (2x +1)(x + 4) (4) x (1) 2x x +2x +1 (4) + F O I L = 2x + 9x + 4 2 = 2x + 8x + x + 4 2 = First Outer Inner Last: FOIL Method. STANDARD MULTIPLYING POLYNOMIALS x x x 1 1 1 1 1 x + 4 2x + 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14. 14 Simplify the following expressions: (2x - 2)(3x - 1) (-1) (3x) (-2) 2x (3x) +2x+(-2) (-1) + F O I L = 6x - 8x + 2 2 = 6x -2x - 6x + 2 2 = 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. 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 1 1 1 x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16. 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 1 1 1 x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17. 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 1 1 1 x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18. 18 Standards 3 and 4 x+1x+1 X +5-3x +5x -3x 2 x 2 x 3 +5 +2x x 3 -2x 2 x- 4x + 7 2 x-4x-4 X -28+16x +7x -4x 2 2 x 3 -28 + 23x x 3 -8x 2 x- 3x + 5 2 2 Simplify x+1 Simplify x- 4x + 7 2 x-4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19. 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. 20 STANDARDS 1 1 1 2 2 2 3 3 3 4 4 4 1x1x1 = 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. 21 x x x x x 1 1 1 1 1 1 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. 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 + 6 2 = x + 3x +2x + 6 2 = x+1x+1 X +6+5x +6x +5x 2 x 2 x 3 +6 +11x x 3 +6x 2 x +5x + 6 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

24. 24 STANDARD Multiply: (x+3)(x+2)(x+1) = x + 6x + 11x + 6 3 2 (x + 2)(x + 3) (3) x (2) x x + x + (2) (3) + F O I L = x + 5x + 6 2 = x + 3x +2x + 6 2 = x+1x+1 X +6+5x +6x +5x 2 x 2 x 3 +6 +11x x 3 +6x 2 x +5x + 6 2 (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. 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 + 3 2 = 2x + 6x +1x + 3 2 = x+4x+4 X +12+28x + 3x +7x 2 8x 2 2x 3 +12 +31x 2x 3 +15x 2 2x +7x + 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26. 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 + 3 2 = 2x + 6x +1x + 3 2 = x+4x+4 X +12+28x + 3x +7x 2 8x 2 x 3 2x +7x + 3 2 +12 +31x 2x 3 +15x 2 +12 +31x 2x 3 +15x 2 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

27. 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 + 3 2 = 2x + 6x +1x + 3 2 = x+4x+4 X +12+28x + 3x +7x 2 8x 2 x 3 2x +7x + 3 2 +12 +31x 2x 3 +15x 2 +12 +31x 2x 3 +15x 2 = (x+3) (x+4) (2x+1) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

28. 28 Standards 3 and 4 x - 4x + 5 2 x - x -7x +15 3 2 x x 3 - 4x 2 + 5x - 3x 2 -12x +15 +3 3x 2 -12x +15 - x -10x + 26 2 x -12x +46x -52 3 2 x x 3 -10x 2 +26x - -2x 2 +20x -52 -2 2x 2 +20x -52 - Divide by x - x -7x +15 3 2 x - 4x + 5 2 Divide by x -10x + 26 2 x -12x +46x -52 3 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

29. 29 Standards 3 and 4 2 1 2 -20 24 1 2 4 -12 8 -24 0 Divide x + 2x -20x + 24 by x-2 using synthetic division 2 3 with x- (+2) x + 4x - 12 1 2 2 -4 1 1 -8 16 1 -4 -3 +4 12 -16 0 Divide x +x - 8x + 16 by x+4 using synthetic division 2 3 with x- (-4) x - 3x + 4 1 2 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

30. 30 Standards 3 and 4 Factoring the Greatest Common Factor (GCF): 4x y z - 16x y z + 32x 3 2 3 2 4 = 4xx y z - 4(4x)xy z + 8(4x) 2 3 2 4 =4x(x y z – 4xy z + 8) 2 3 2 4 -27p q r + 9p q r - 3pqr 3 3 2 2 2 3 = (3)(-9)pp qq r + (3)(3)ppqqrr -3pqrr 2 2 2 =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. 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 ) 3 3 2 2 8y + 27z = 33 64k +125j = 33 Difference of Two Cubes: a - b = (a-b)(a +ab + b ) 3 3 2 2 216y - 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. 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. 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. 34 Difference of Two Squares: (x+2)(x-2) = x - 4 2 STANDARD SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

35. 35 STANDARD x x 1 1 1 x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

36. 36 STANDARD x x 1 1 1 x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

37. 37 STANDARD x x 1 1 1 x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

38. 38 STANDARD x x 1 1 1 x + 3 x – 3 (x+3)(x-3) = x – 9 2 SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

39. 39 Standards 3 and 4 Perfect Square Trinomials: a + 2ab + b = (a + b) 2 2 2 a - 2ab + b = (a - b) 2 2 2 x + 4x + 4 2 = (x +2) 2 x + 6x + 9 2 = (x +3) 2 = (x) + 2(x)(2) + (2) 2 2 = (x) + 2(x)(3) + (3) 2 2 25x + 40x + 16 2 = (5x) + 2(5x)(4) + (4) 2 2 = (5x + 4) 2 x -10x + 25 2 = (x - 5) 2 x - 14x +49 2 = (x -7) 2 = (x) - 2(x)(5) + (5) 2 2 = (x) - 2(x)(7) + (7) 2 2 64x - 64x + 16 2 = (8x) - 2(8x)(4) + (4) 2 2 = (8x - 4) 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

41. 41 STANDARD SPECIAL PRODUCTS (x +3) 2 = (x) + 2(x)(3) + (3) 2 2 x + 6x + 9 2 = x x 1 1 1 x +3 1 = (x+3)(x+3) 1 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

42. 42 Standards 3 and 4 General Trinomials: B -5B -50 2 (B+5) (B-10) -5 -50 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)= -23 1 x Factor the following trinomial: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

43. 43 Standards 3 and 4 -60 -11 x (4)(-15) 4 + -15= -11 12x - 11x -5 2 12x + (4-15)x -5 2 12x + 4x -15x -5 2 4x(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) 3 + -20= -17 (2)(-30) 2 + -30= -28 (1)(-60) 1 + -60= -59 12x - 11x -5 2 General Trinomials: acx + (ad + bc)x + bd = (ax +b)(cx +d) 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

44. 44 Standards 3 and 4 +24 +11 x -6x +11x -4 2 -6x + (3+8)x -4 2 -6x + 3x +8x -4 2 -3x(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) 2 + 12= 14 (1)(24) 1 + 24= 25 -6x +11x - 4 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

45. 45 Standards 3 and 4 -96 -4 x (8)(-12) 8 + -12= -4 8x - 4x -12 2 8x + (8-12)x -12 2 8x + 8x -12x -12 2 2x(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 -12 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved