The Distributive Property Basic College Math The Distributive Property

Getting ready for algebra! Now that we’re ready to study algebra, we’re going to need more than just the order of operations to keep us in line!

The distributive property. A brand new property! We’re going to need a new property called the distributive property of multiplication over addition. It also has a sister called the distributive property of multiplication over subtraction. Collectively, they are sometimes just called The distributive property.

What will we learn? We are going to see how the property works with a numerical example. We’re going to see how it works with some algebraic examples. We’re going to map out the potential pitfalls of working with this property. We’re going to read a story to help us remember how to use it correctly!

Remember the order of operations? If you have a problem like 5( 6 + 8 ), the order of operations tells you to do what is in parentheses first. 5( 6 + 8 ) becomes 5( 14 ) which equals 70. Remember: the 5 directly against the open parentheses means to multiply by 5.

The Distributive Property of Multiplication over Addition tells us . . . If we have something that is multiplying the sum of two or more numbers like 5( 6 + 8 ). . . We have an alternate way to calculate the answer.

Practical Example Suppose you hire the neighbor’s daughter for $5.00 an hour to help you rake leaves. She is all excited about the job because she wants to buy a stereo for her room.

Raking in the profits WOW! Day 1, she works 6 hours that’s $5(6) or $30 toward her stereo. Day 2, she works 8 hours that’s another $5(8) or $40 toward her stereo. WOW! $30 + $40 equals $70 for her stereo!

Paying the bill You go to the bank to get the cash to pay her. All you care about is that she worked 14 hours. $5(14) = $70. You both get the same answer even though you did the problem differently. The Distributive Property tells you that it is OK. You can multiply and then add (like she did) or add and then multiply (like you did).

Some algebraic examples 5( x + 2) = 5( x ) + 5( +2 ) = 5x + 10 2( 3x – 6 ) = 2( 3x ) + 2( -6 ) = 6x - 12

A more complicated problem Take the problem - 5( x + 7 ) + 30 You first distribute and then combine like terms. Answer: - 5( x + 7 ) + 30 = - 5( x ) - 5( 7 ) + 30 = - 5x - 35 + 30 = - 5x - 5

Pitfall 1: Negative numbers If the number in front of the parentheses is negative, you must multiply each item in the parentheses by the negative number. Watch your signs! - 6(x – 4) = - 6( x ) - 6( - 4 ) = - 6x + 24 -2(2x + 5) = -2( 2x ) - 2( + 5 ) = -4x -10 -3(-3x – 4) = -3( - 3x) - 3(- 4) = 9x + 12

Pitfall 2: Oops - forgot the second guy! 5( x + 7 ) = 5x + 7 WRONG! You get the 5x right because it’s staring you in the face. In your haste to get the right answer, you forget all about the poor 7. 5(x + 7) = 5( x ) + 5( 7 ) = 5x + 35 RIGHT!

Pitfall 3: Over enthusiasm 5(x + 7) – 30 = 5( x ) + 5( 7 ) – 5( 30 ) 5x + 35 – 150 = 5x - 115 WRONG You get so excited about distributing the 5 that you distribute him to everybody – even the numbers that aren’t in the parentheses! 5( x + 7 ) – 30 = 5( x ) + 5( 7 ) – 30 5x + 35 – 30 = 5x + 5 RIGHT!

Pitfall 4: It’s backwards! ( x + 7 )(5) What the heck? The commutative property of multiplication tells us that we can re-write the problem because order doesn’t matter when you multiply. 5( x + 7 ) Ah, that’s better!

Let me tell you a story . . . The first time I ever went to Sea World, my friends and I decided to go to the Shamu show. When we entered the area, I noticed that there were seats right down at the front that were painted red instead of blue, like the other seats. There were no “Reserved” signs on them, so we decided to go sit there and move if we were told we had to.

THE SEATS WERE GREAT! We could see Shamu swimming around through the windows in the side of his tank. At last, the show started and Shamu did a mighty leap out of the water right in front of us. It was amazing! I had my camera out, ready to shoot the picture of a lifetime. . .

. . . and then he re-entered the water . . . Perfect! . . . and then he re-entered the water . . .

. . . and that is how Shamu can help you remember the rules. When you are simplifying an expression like 5 ( x + 7 ) – 30 The 30 is in the blue seats – high and dry and out of harm’s way. The x and the 7 are in the red seats – definitely the splash zone! And the 5 is Shamu!

Splash photo courtesy of Mike Sandells www.mikejs.com The End Splash photo courtesy of Mike Sandells www.mikejs.com