chapter 01 – section 07

Chapter 01 – Section 07

The Distributive Property

© William James Calhoun

To use the distributive property to simplify expressions.

Look at this problem:2(4 + 3)

Through your knowledge of order of operations, you know what to do first to evaluate this expression.

2(7)14

Now, look what happens when I do something different with the problem.

2(4 + 3) = 8 6+ = 14No difference.This is an example of the distributive property.

* *

© William James Calhoun

Now why would one ever use the distributive property to solve 2(4 + 3)?

The answer is generally,

“Never! Just use the order of operations.”

Where this is going to become very important is when we have an expression in the parenthesis which can not be simplified, like:

2(4 + x)

You need to be able to recognize and use the distributive property throughout all of Algebra.

This is the one property you need to know by name, forwards, and backwards!

© William James Calhoun

For any numbers a, b, and c,a(b + c) = ab + ac and (b + c)a = ba + ca;a(b - c) = ab - ac and (b - c)a = ba - ca.

1.7.1 DISTRIBUTIVE PROPERTY

Another way to think of it is, “When multiplying into parenthesis, everything on the inside gets a piece of what is on the outside.”

Notice the number to be distributed can either be at the front of the parenthesis or at the back.

If there is no number visible in front or in back of the parenthesis, the number to be distributed is 1.

© William James Calhoun

Here is another way to look at the Distributive Property.

You have two squares. One is 3ft by 5ft. The second is 3ft by 8ft.

You want to know the total area of the two squares. Set them side by side.

Now, you can either:1) Add the continuous length, then multiply by the width; or2) Multiply out the area of each box, then add (distribution)

3 * 5 = 15 3 * 8 = 24 15 + 24 = 39 square ft

5 + 8 = 13

3 13 * 3 = 39 square ft

© William James Calhoun

SPECIAL NOTE:

The first two examples in the book force you to use the distributive property when it is not necessary - AND - contrary to the order of operations rules we have gone over.

For distribution problems that have no variables in them – simply use the order of operation.

The book uses the non-variable distribution problems to prove that distribution works – but you already know that by now!

Taking extra steps is not very helpful, but here is one of those examples.

© William James Calhoun

EXAMPLE 1α: Use the distributive property to find each product.a. 7 * 98 b. 8(6.5)

EX1EX1β

The book would have you break this problem down into:

7(100 – 2)

Then distribute.

700 – 14

Finally, subtract.

686

The book would have you break this problem down into:

8(6 + 0.5)

Then distribute.

48 + 4

Finally, add.

52There is some merit to part B…that is a good way to solve the problem without a calculator.With a calculator available, however, why bother distributing?

© William James Calhoun

EXAMPLE 1β: Use the distributive property (if necessary) to find each product.

a. 16(101) b. 9(10.6)

© William James Calhoun

Here a a couple of definitions that will be used a great deal.

• term - number, variable, or product or quotient of numbers and variables

Examples of terms: x3, 1/4a, and 4y.

The expression 9y2 + 13y + 3 has three terms.

• like terms - terms that contain the same variables, with corresponding variables having the same power

In other words, the terms have the exact same letter configuration.

TERMS

© William James Calhoun

SPECIAL NOTE:

Terms must have the EXACT same letters to the EXACT same powers in order to be LIKE terms!

In the expression 8x2 + 2x2 + 5a + a, 8x2 and 2x2 are like terms.

5a and a are also like terms.

Another way to think of it is this:

Like terms are alike in that they have the exact same letter configuration.

8x and 4x2 are not like terms because the x’s are not the same as the x’s-squared.

© William James Calhoun

ONLY LIKE TERMS CAN BE COMBINED THROUGH ADDITION AND SUBTRACTION.

Since 3x and 8x are like terms, they can combine - both have the same letter configuration - an “x” to the 1st power.

We can use the distributive property to undistribute the x and combine the numbers:

3x + 8x= (3 + 8)x= 11x

Another way to look at the problem is, “You have three x’s plus eight x’s. All told, how many x’s do you have?”

The answer is, “You have eleven x’s,” or just:11x.

© William James Calhoun

To simplify an expression in math, you must:1) Have all like terms combined; and2) Have NO parenthesis are present.

EXAMPLE 2α: Simplify

EX2EX2β

2 2 21 11x 2x x .4 4

In this expression, are all like terms.

Undistribute the x2.

Add the numbers up.

Slap the x2 onto the number.

= (5)x2

= 5x2

2 2 21 11x , 2x , and x4 4

21 112 x4 4

2 2 21 11x 2x x4 4

© William James Calhoun

EXAMPLE 2β: Simplify 2 26xy 7xy .

5 5

© William James Calhoun

EXAMPLE 3α: Name the coefficient in each term.a. 145x2y b. ab2 c.

• coefficient - the number in front of the letters in a term

In the term 23ab, 23 is the coefficient.

In xy, the coefficient is 1.

NEVER FORGET THE “INVISIBLE” ONE!

24a5

145 1 4/5

EXAMPLE 3β: Name the coefficient in each term.a. y2 b. c.

r4

25x7

© William James Calhoun

EXAMPLE 4α: Simplify each expression.a. 4w4 + w4 + 3w2 - 2w2 b.

33a 2a

4

EXAMPLE 4β: Simplify each expression.a. 13a2 + 8a2 + 6b b. 25m m 4m

6 6

= (4 + 1)w4 + (3 - 2)w2

= 5w4 + 1w2

= 5w4 + w2

= (1/4 + 2)a3

= 21/4a3

© William James Calhoun

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