1 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 1 STANDARD 10 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 2 POLYNOMIALS MULTIPLY WITH DISTRIBUTIVE PROPERTY:

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Presentation transcript:

1 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 1 STANDARD 10 MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 2 POLYNOMIALS MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 3 USE DISTRIBUTIVE PROPERTY TO MULTIPLY BINOMIALS: Problems MULTIPLY BINOMIALS WITH F.O.I.L.: Problems and MODELING MULTIPLY POLYNOMIALS VERTICAL FORMAT: Problems DIFFERENCE OF SQUARES: Problems and MODELING PERFECT SQUARE TRINOMIALS: Problems and MODELING FACTORING GENERAL TRINOMIALS MODELING THIRD DEGREE POLYNOMIALS END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 10: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. ESTÁNDAR 10: Los estudiantes suman restan, multiplican, y dividen monomios y polinomios. Los estudiantes resuelven problemas de múltiples pasos, incluyendo problemas escritos, usando estas técnicas. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 It is possible to use the distributive property of multiplication over addition to multiply polynomials: STANDARD 10 Simplify 5y(6y + 5) =5y(6y) + 5y(5) 5y(6y + 5) =30y + 25y 2 MULTIPLYING POLYNOMIALS Simplify 6p(7p + 5) =6p(7p) + 6p(5) 6p(7p + 5) =42p + 30p 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 It is possible to use the distributive property of multiplication over addition to multiply polynomials: STANDARD 10 Simplify -8x(2x + 3x – 6) 2 -8x(2x + 3x – 6) 2 = -8x(2x ) + (-8x)(3x) + (-8x)(-6) 2 = -16x - 24x + 48x 3 2 MULTIPLYING POLYNOMIALS = (-8)(2)x + (-8)(3)x + (-8)(-6)x Simplify -5x(3x + 2x – 3) 2 -5x(3x + 2x – 3) 2 = -5x(3x ) + (-5x)(2x) + (-5x)(-3) 2 = -15x - 10x + 15x 3 2 = (-5)(3)x + (-5)(2)x + (-5)(-3)x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 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 STANDARD 10 MULTIPLYING POLYNOMIALS Simplify 3x(4x y + 5x y – 7x y ) = (3x)(4x y) + (3x)(5x y ) + (3x)(-7x y ) x(4x y + 5x y – 7x y ) = (3)(4)x y + (3)(5)x y + (3)(-7)x y =12x y + 15x y -21x y PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 = (2x +1)(x + 4) (4)x (1) 2x x +2x +1(4) + = 2x + 9x = 2x + 8x + x = = (6x +3)(2x + 5) (5) (2x) (3) 6x (2x) +6x +3 (5) + =12x + 36x =12x + 30x + 6x = (6x - 3)(x + 5) (5) x (-3) 6x x +6x +(-3)(5) + = 6x + 27x = 6x + 30x -3x = (4x - 3)(3x - 7) (-7) (3x) (-3) 4x (3x) +4x +(-3) (-7) + =12x - 37x =12x -28x - 9x = Multiply the following binomials USING THE DISTRIBUTIVE PROPERTY: STANDARD 10 MULTIPLYING POLYNOMIALS = 2x(x+4) + 1(x+4) = 6x(2x+5) + 3(2x+5) (2x +1)(x + 4) (6x +3)(2x + 5) = (6x - 3)(x + 5) = 6x(x+5) + (-3)(x+5) = (4x - 3)(3x - 7) = 4x(3x-7) + (-3)(3x-7) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 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. STANDARD 10 MULTIPLYING POLYNOMIALS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 (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 10 MULTIPLYING POLYNOMIALS x x x x + 4 2x + 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 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 10 MULTIPLYING POLYNOMIALS x x x 3x – 1 2x – 2 x x PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 (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 10 MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 (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 10 MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 (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 10 MULTIPLYING POLYNOMIALS x x x x + 3 2x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 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 x+1 x -4x x-4 STANDARD 10 MULTIPLYING POLYNOMIALS MULTIPLY VERTICALY: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 x-2x-2 X - 8+4x -2x 2 2 x x x 3 -4x 2 x - 3x x+3x+3 X x +6x -3x 2 3x 2 x x x 3 +0x 2 x -2x x-2 x -3x x+3 STANDARD 10 MULTIPLYING POLYNOMIALS MULTIPLY VERTICALY: x x 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 Difference of Two Squares: (x+2)(x-2) STANDARD 10 SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 Difference of Two Squares: (x+2)(x-2) STANDARD 10 SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 Difference of Two Squares: (x+2)(x-2) = x STANDARD 10 SPECIAL PRODUCTS x x 1 1 x +2 x – 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 STANDARD 10 x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 STANDARD 10 x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 STANDARD 10 x x x + 3 x – 3 (x+3)(x-3) SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 STANDARD 10 x x x + 3 x – 3 (x+3)(x-3) = x – 9 2 SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 Difference of Two Squares: (x+2)(x-2) = x (a+b)(a-b) = a - b 2 2 (3y+8)(3y-8) STANDARD 10 9y = (p+4)(p-4) = p (2y+5)(2y-5) 4y = SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 STANDARD 10 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

24 STANDARD 10 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

25 STANDARD 10 Perfect Square Trinomials: (a – b) = a - 2ab + b (a + b) = a + 2ab + b (x +2) 2 (x +3) 2 = (x) + 2(x)(2) + (2) 2 2 = (x) + 2(x)(3) + (3) 2 2 = (5x) + 2(5x)(4) + (4) 2 2 (5x + 4) 2 (x - 5) 2 (x -7) 2 = (x) - 2(x)(5) + (5) 2 2 = (x) - 2(x)(7) + (7) 2 2 = (8x) - 2(8x)(4) + (4) 2 2 (8x - 4) 2 x + 4x = x - 14x = 25x + 40x = x -10x = x + 6x = 64x - 64x = SPECIAL PRODUCTS PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26 STANDARD 10 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

27 STANDARD 10 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

28 STANDARD 10 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

29 STANDARD 10 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

30 STANDARD 10 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

31 STANDARD 10 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

32 General Trinomials: acx + (ad + bc)x + bd = (ax +b)(cx +d) 2 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: STANDARD 11 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARD 11 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARD 11 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARD 11 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved