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

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MULTIPLY WITH DISTRIBUTIVE PROPERTY: Problems 1

STANDARD 10


POLYNOMIALS


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 SHOWPRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

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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 + 25y2

MULTIPLYING POLYNOMIALS

Simplify 6p(7p + 5)

=6p(7p) + 6p(5)6p(7p + 5)

=42p + 30p2



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STANDARD 10

Simplify -8x(2x + 3x – 6)2

-8x(2x + 3x – 6)2 = -8x(2x ) + (-8x)(3x) + (-8x)(-6)2

= -16x - 24x + 48x3 2

MULTIPLYING POLYNOMIALS

= (-8)(2)x + (-8)(3)x + (-8)(-6)x2+1 1+1

Simplify -5x(3x + 2x – 3)2

-5x(3x + 2x – 3)2 = -5x(3x ) + (-5x)(2x) + (-5x)(-3)2

= -15x - 10x + 15x3 2

= (-5)(3)x + (-5)(2)x + (-5)(-3)x2+1 1+1



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Simplify 4x(2x y + 3x y – 6x y )5 4 3 32

= (4x)(2x y) + (4x)(3x y ) + (4x)(-6x y )5 4 2 3 34x(2x y + 3x y – 6x y )5 4 3 32

= (4)(2)x y + (4)(3)x y + (4)(-6)x y1+5 1+4 1+32 3

=8x y + 12x y -24x y6 5 2 4 3

STANDARD 10MULTIPLYING POLYNOMIALS

Simplify 3x(4x y + 5x y – 7x y )6 4 3 32

= (3x)(4x y) + (3x)(5x y ) + (3x)(-7x y )6 4 2 3 33x(4x y + 5x y – 7x y )6 4 3 32

= (3)(4)x y + (3)(5)x y + (3)(-7)x y1+6 1+4 1+32 3

=12x y + 15x y -21x y7 5 2 4 3



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= (2x +1)(x + 4)

(4) x (1)2x x+2x

+1

(4)+

= 2x + 9x + 42

= 2x + 8x + x + 42

=

= (6x +3)(2x + 5)

(5) (2x) (3)6x (2x)+6x

+3

(5)+

=12x + 36x + 152

=12x + 30x + 6x + 152

=

(6x - 3)(x + 5)

(5) x (-3)6x x+6x

+(-3)

(5)+

= 6x + 27x - 152

= 6x + 30x -3x - 152

=

(4x - 3)(3x - 7)

(-7) (3x) (-3)4x (3x) +4x

+(-3)

(-7)+

=12x - 37x + 212

=12x -28x - 9x + 212

=

Multiply the following binomials USING THE DISTRIBUTIVE PROPERTY:


= 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)



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Simplify the following expressions:

(6x +3)(2x + 5)

(5) (2x) (3)6x (2x)+6x

+3

(5)+

F O I L

=12x + 36x + 152

=12x + 30x + 6x + 152

=

(6x - 3)(x + 5)

(5) x (-3)6x x+6x

+(-3)

(5)+F O I L

= 6x + 27x - 152

= 6x + 30x -3x - 152

=

(4x - 3)(3x - 7)

(-7) (3x) (-3)4x (3x) +4x

+(-3)

(-7)+

F O I L

=12x - 37x + 212

=12x -28x - 9x + 212

=

First Outer Inner Last: FOIL Method.




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(2x +1)(x + 4)

(4) x (1)2x x+2x

+1

(4)+F O I L

= 2x + 9x + 42

= 2x + 8x + x + 42

=



x

x

x

1

1 1 1 1

x + 4

2x + 1



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Simplify the following expressions:

(2x - 2)(3x - 1)

(-1) (3x) (-2)2x (3x)+2x

+(-2)

(-1)+

F O I L

= 6x - 8x + 22

= 6x -2x - 6x + 22

=



x

x

x

-1

-1

3x – 1

2x – 2

-1

x x



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(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

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



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(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

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



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(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

=



x

x

x

-1

1 1 1

x + 3

2x – 2

-1



13

x+1X

+5-3x+5x-3x2

x2

x3

+5+2xx3 -2x2

x- 4x + 7

2

x-4X

-28+16x+7x-4x2

-4x2

x3

-28+ 23xx3-8x2

x- 3x + 5

2

x -3x + 5

2

x+1 x -4x + 7

2

x-4

STANDARD 10MULTIPLYING POLYNOMIALSMULTIPLY VERTICALY:



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x-2X

- 8+4x+4x-2x2

- 2x2

x3

- 8+8xx3 -4x2

x - 3x + 6

2

x+3X

+18- 9x+6x-3x2

3x2

x3

+18 - 3xx3+0x2

x -2x + 4

2

x -2x + 4

2

x-2 x -3x + 6

2

x+3

STANDARD 10MULTIPLYING POLYNOMIALSMULTIPLY VERTICALY:

+18 - 3xx3



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Difference of Two Squares:

(x+2)(x-2)

STANDARD 10SPECIAL PRODUCTS

x

x

-1

1 1

x +2

x – 2

-1



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(x+2)(x-2)


x

x

-1

1 1

x +2

x – 2

-1



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(x+2)(x-2) = x - 4 2


x

x

-1

1 1

x +2

x – 2

-1



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STANDARD 10

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



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STANDARD 10

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



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STANDARD 10

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3)

SPECIAL PRODUCTS



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STANDARD 10

x

x 1 1 1

x + 3

x – 3 -1

-1

-1

(x+3)(x-3) = x – 9

2

SPECIAL PRODUCTS



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(x+2)(x-2) = x - 4 2

(a+b)(a-b) = a - b2 2

(3y+8)(3y-8)

STANDARD 10

9y - 642=

(p+4)(p-4) = p - 162

(2y+5)(2y-5) 4y - 252=

SPECIAL PRODUCTS



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(x +2)2

= (x) + 2(x)(2) + (2)2 2

x + 4x + 42=

x

x

1

1 1

x +2

x + 2

1

= (x+2)(x+2)



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(x +3)2

= (x) + 2(x)(3) + (3)2 2

x + 6x + 92=

x

x

1

1 1

x +3

x + 3

1

= (x+3)(x+3)

1

1



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STANDARD 10

Perfect Square Trinomials:

(a – b) = a - 2ab + b222

(a + b) = a + 2ab + b222

(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 + 42=

x - 14x +492=

25x + 40x + 162=

x -10x + 252=

x + 6x + 92=

64x - 64x + 162=

SPECIAL PRODUCTS



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STANDARD 10

Multiply:

(x+3)(x+2)(x+1)

(x+2)

(x+1)

(x+3)

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2



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STANDARD 10

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

x +5x + 6

2



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STANDARD 10

Multiply:

(x+3)(x+2)(x+1) = x + 6x + 11x + 63 2

(x + 2)(x + 3)

(3) x (2) xx+ x

+ (2)

(3)+F O I L

= x + 5x + 6 2

= x + 3x +2x + 6 2

=

x+1X

+6+5x+6x+5x 2

x2

x3

+6+11xx3 +6x2

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).



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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 + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

2x 3

+12+31x2x 3+15x 2

2x +7x + 3

2


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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 + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=


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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 + 3 2

= 2x + 6x +1x + 3 2

=

x+4X

+12+28x+ 3x+7x 2

8x2

x3

2x +7x + 3

2

+12+31x2x 3+15x 2

+12+31x2x 3+15x 2=

(x+3)

(x+4)

(2x+1)


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General Trinomials:

acx + (ad + bc)x + bd = (ax +b)(cx +d)2

B -5B -502

(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

XFactor the following trinomial:

STANDARD 11


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-60 -11

X

(4)(-15) 4 + -15= -11

12x - 11x -52

12x + (4-15)x -52

12x + 4x -15x -52

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 -52

General Trinomials:

acx + (ad + bc)x + bd = (ax +b)(cx +d)2

STANDARD 11


34

+24 +11

X

-6x +11x -42

-6x + (3+8)x -42

-6x + 3x +8x -42

-3x(2x)- (-3x)1 +4(2x) + (4)(-1)

-3x(2x-1) + 4(2x -1)

(-3x+ 4)(2x-1)


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 - 42

STANDARD 11


35

-96 -4

X

(8)(-12) 8 + -12= -4

8x - 4x -122

8x + (8-12)x -122

8x + 8x -12x -122

2x(4x)+ (2x)4 -3(4x) + (-3)(4)

2x(4x+4) – 3 (4x +4)

(2x- 3)(4x+4)


Find two numbers that multiplied be (8)(-12)=-96 and added -4.

8x - 4x -122

STANDARD 11


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

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