Multiplying and Dividing an Algebraic Expression 1 G

CHAPTER 8 Multiplying and Dividing C an Algebraic Expression

c GOAL Multiply or divide an algebraic expression by a constant.

Learn about the Math

Jorge is purchasing wood to build a stage for the school play. The stage is made up of three identical rectangular sections. He is told that the area of each section in square metres can be described by the algebraic expression 2x2 1 3x 1 1. Jorge must calculate the total area of all three sections. ? How can Jorge use the algebraic expression 2x2 1 3x 1 1 to calculate the total area of all three sections of the stage?

A. Draw a diagram representing the three identical rectangular sections of the stage. B. Label each rectangle with the expression 2x2 1 3x 1 1. C. Since all three sections of the stage are identical, multiplication can be used to find the expression that would represent the total area of all three. Write a multiplication expression showing 2x2 1 3x 1 1 being multiplied by 3. D. Now use the distributive property to distribute the 3 over all three terms of the expression. E. Multiply each coefficient or constant by 3 and leave the variables unchanged. F. Remove the bracket to form an expression in simplest form that describes the total area of all three sections of the stage.

Copyright © 2009 by Nelson Education Ltd. Reproduction permitted for classrooms 8C Multiplying and Dividing an Algebraic Expression 1 G. Jorge is told that if he replaces the x terms by 1, he will be able to calculate the actual area of the stage as a numerical value. Substitute 1 into your expression for x. H. Simplify the expression. The solution will be equal to the total area of the three sections of the stage, in square metres.

Reflecting

1. If an expression has three terms, how many terms will it have after being multiplied by a constant? Explain. 2. If instead of 3, you were to multiply the expression by 23 , how would that change your answer? Explain. 3. How would you expect dividing an expression by a constant to differ from multiplying it by a constant?

Work with the Math

Example 1: Multiplying an algebraic expression by a constant

Multiply 25(2x3 1 6x2 2 8x). Akeem’s Solution

(25 3 2x3) 1 (25 3 6x2) 2 (25 3 8x) First, I must use the distributive property to distribute the 25 over all three terms in this expression. (210x3) 1 (230x2) 2 (240x) Then, I multiply each coefficient by 25 and leave the variables unchanged. 210x3 2 30x2 1 40x Finally, I remove the bracket and write in simplest form. 25(2x3 1 6x2 2 8x) 5210x3 2 30x2 1 40x

Divide (10x3 1 12x2 2 6x 1 2) by 2. Denise’s Solution 10 12 6 2 x3 1 x2 2 x 1 I must rewrite the expression, dividing every 2 2 2 2 coefficient or constant by the divisor, 2. 5x3 1 6x2 2 3x 1 1 Then, I use division to simplify the coefficients and constants and leave the variables unchanged. (10x3 1 12x2 2 6x 1 2) 4 2 5 5x3 1 6x2 2 3x 1 1

A Checking C Extending 4. Multiply 10(3x2 1 7x 1 7). 8. Write each expression in simplest 5. Divide (9x3 1 18x2 1 12x) by 3. form. a) 2 B Practising 2(10x 1 15) 4 10 b) 4 3 2 6. Distribute the constant over each term 25(12x 1 7x 1 28x 1 24x) 4 5 c) 2 2 in the following expressions. 7(x 1 11 1 8x) 1 3(x 2 4) d) 3(3x2 1 6x 2 9) 4 9 2 a) 8(12x 2 3) e) 2(2(2(x 1 1))) 4 8 3 b) 11(14x 1 8x 1 5) f) 2(x2 1 1) 2 3(x 1 1) c) 220(10x2 2 9x 1 22) 9. A rectangle has dimensions 3 m by d) 15(x4 1 5x3 2 15x2 1 x 2 1) 2 2 2x 1 4x 1 8 metres. What is the area e) 24(100x 1 44x 1 5) of each of the triangles formed by 7. Find each product or quotient. Write drawing one diagonal of the rectangle? your answers in simplest form. 10. If c and d are whole numbers 2 a) 12(3x 1 6x 2 4) and neither c nor d is 0, what 3 b) 27(11x 2 2x 1 18) must be true about c and d if the 2 c) (12x 1 4x 1 16) 4 4 coefficients of c(2x2 1 3x 1 1) 4 d 4 3 2 d) 24(22x 2 3x 1 4x 1 2x 2 20) are whole numbers? e) (25x2 1 10x 2 5) 4 (25) f) 20(10x3 2 5x 1 11) g) (35x2 1 28x 2 14) 4 7 h) (54x4 2 36x3 2 27x2 1 9x 2 18) 4 9 i) (62x2 1 2x 2 4) 4 (22) j) 28(5x4 1 16x2 1 8x 2 1) k) 88(10x2 2 9x) l) (50x2 2 30x 1 40) 4 10 m) (121x2 1 11x) 4 11