Ch 2.6

Objective:

To use the distributive property to simplify variable expressions.

PropertyDistributive Property

The distributive property is used when multiplying an expression with a group of expressions that are added (or subtracted).

For example: a(b + c) = a(b) + a(c)

a(b - c) = a(b) - a(c)

 

(b + c)a = (b)a + (c)a

(b - c)a = (b)a - (c)a

THE DISTRIBUTIVE PROPERTY

a(b + c) = ab + ac

(b + c)a = ba + ca

2(x + 5) 2(x) + 2(5) 2x + 10

(x + 5)2 (x)2 + (5)2 2x + 10

(1 + 5x)2(1)2 + (5x)22 + 10x

y(1 – y)y(1) – y(y)y – y

2

=

=

==

=

=

==

The product of a and (b + c):

USE THE DISTRIBUTIVE PROPERTY

Comparison

Order of Operations Distributive Property

6(3 + 5) 6(3 + 5)

6(8)

48

6(3) + 6(5)

18 + 30

48

Why distribute when order of operations is faster ?

Use Distributive Property when there is a variable

Use Order of Operation to “check” your answer

Use the distributive property to simplify.

1) 3(x + 7)

2) 2(a - 4)

3) -7(8 - m)

4) 3(4 - a)

5) (3 - k)5

6) x(a + m)

7) -4(3 - r)

8) 2(x - 8)

9) -1(2m - 3)

10) (6 - 2y)3

3x + 21

2a - 8

-56 + 7m

12 - 3a

15 - 5k

ax + mx

-12 + 4r

2x - 16

-2m + 3

18 - 6y

(y – 5)(–2)= (y)(–2) + (–5)(–2)

= –2y + 10

– (7 – 3x)= (–1)(7) + (–1)(–3x)

= –7 + 3x

= –3 – 3x

(–3)(1 + x)

= (–3)(1) + (–3)(x)

USE THE DISTRIBUTIVE PROPERTY

Remember that a factor must multiply EACH term of an expression.

Forgetting to distribute the negative sign when multiplying by a negative factor is a common error.

Use the distributive property to simplify.

1) 4(y - 7)

2) 3(b + 4)

3) -5(9 - m)

4) 5(4 - a)

5) (7 - k)6

6) a(c + d)

7) - (-3 - r)

8) 4(x - 8)

9) - (2m + 3)

10) (6 - 2y) -3y

4y - 28

3b + 12

-45 + 5m

20 - 5a

42 - 6k

ac + ad

3 + r

4x - 32

-2m - 3

6 - 2y -3y

Find the difference mentally.

Find the products mentally.

The mental math is easier if you think of $11.95 as $12.00 – $.05.

Write 11.95 as a difference.

You are shopping for CDs.You want to buy six CDs

for $11.95 each.

Use the distributive propertyto calculate the total cost

mentally.

6(11.95) = 6(12 – 0.05) Use the distributive property.

= 6(12) – 6(0.05)

= 72 – 0.30

= 71.70

The total cost of 6 CDs at $11.95 each is $71.70.

MENTAL MATH CALCULATIONS

Combine like terms.

SIMPLIFYING BY COMBINING LIKE TERMS

4x2 + 2 – x2 =

(8 + 3)x Use the distributive property.

= 11x Add coefficients.

8x + 3x

Group like terms.

Rewrite as addition expression.

Distribute the –2.

Multiply.

Combine like terms and simplify

4x2 – x2 + 2 = 3x2 + 2

3 – 2(4 + x) = 3 + (–2)(4 + x)

= 3 + [(–2)(4) + (–2)(x)]

= 3 + (–8) + (–2x)

= –5 + (–2x) = –5 – 2x

=

Designate one sign in front of 2x

Subtracting a Quantity

1) -(x + 6)

2) -(2x - 8)

3) 10- (4m + 3)

4) 2(x - 5) - (x - 3)

5) -(3a + 1)

6) -(-3x + 2x -7)

7) -12 - (3y - 8)

8) 4(3k - 5) - (2k + 9)

-x - 6

-2x + 8

10 - 4m - 3

- 4m + 7

2x - 10 - x + 3 x - 7

-3a - 1

+3x - 2x + 7

-12 - 3y + 8

- 3y - 4

12k - 20 - 2k - 9 10k - 29

2

2

Geometric Model for Area3 + 7

4

Two ways to find the total area.

Width by total length (Order of Operations)

Sum of smaller rectangles (Distributive Property)

4(3 + 7) 4(3) + 4(7)

4(3) 4(7)

=4 (10) = 12 + 2840 = 40

Geometric Model for Distributive Property4 x

9

Two ways to find the total area.

Width by total length Sum of smaller rectangles

9(4 + x) 9(4) + 9(x)=