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) = 86+=14 No 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 will use the distributive property throughout all of Algebra.

© 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. 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 are 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: x 3, 1 / 4 a, and 4y. The expression 9y y + 3 has three terms. LIKE TERMS - terms that contain the same variables, with corresponding variables having the same power TERMS

© William James Calhoun Terms must have the EXACT same letters to the EXACT same powers in order to be LIKE terms! 8x 2 + 2x 2 + 5a + a 8x 2 and 2x 2 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 4x 2 are not like terms because they do not have the same power.

© 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; and 2) Have NO parenthesis are present. EXAMPLE 2α: Simplify In this expression, are all like terms. Undistribute the x 2. Add the numbers up. Final answer. = (5)x 2 = 5x 2

© William James Calhoun EXAMPLE: Name the coefficient in each term. a. 145x 2 y b. ab 2 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! / 5 EXAMPLE: Name the coefficient in each term. a. y 2 b. c.

© William James Calhoun EXAMPLE 4α: Simplify each expression. a. 4w 4 + w 4 + 3w 2 - 2w 2 b. EXAMPLE: Simplify each expression. a. 13a 2 + 8a 2 + 6bb. = (4 + 1)w 4 + (3 - 2)w 2 = 5w 4 + 1w 2 = 5w 4 + w 2 = ( 1 / 4 + 2)a 3 = 2 1 / 4 a 3