Download Multiplying and Dividing an Algebraic Expression

CHAPTER 8
C
Multiplying and Dividing
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.
can Jorge use the algebraic expression
? How
2x 1 3x 1 1 to calculate the total area of all three
2
sections of the stage?
A. Draw a diagram representing the three identical rectangular
sections of the stage.
B. Label each rectangle with the expression 2x 2 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 2x 2 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.
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8C Multiplying and Dividing an Algebraic Expression
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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 2x 3 ) 1 (25 3 6x 2 ) 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) 5 210x3 2 30x2 1 40x
2
Nelson Mathematics Secondary Year Two, Cycle One
Reproduction permitted for classrooms
Copyright © 2009 by Nelson Education Ltd.
Example 2: Dividing an algebraic expression by a constant
Divide (10x3 1 12x2 2 6x 1 2) by 2.
Denise’s Solution
10 3 12 2 6
2
x 1
x 2 x1
2
2
2
2
5x 3 1 6x 2 2 3x 1 1
I must rewrite the expression, dividing every
coefficient or constant by the divisor, 2.
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
Checking
A
C
2
4. Multiply 10(3x 1 7x 1 7).
5. Divide (9x3 1 18x2 1 12x) by 3.
B
Practising
6. Distribute the constant over each term
in the following expressions.
a)
b)
c)
d)
e)
8(12x2 2 3)
11(14x3 1 8x 1 5)
220(10x2 2 9x 1 22)
15(x4 1 5x3 2 15x2 1 x 2 1)
24(100x2 1 44x 1 5)
7. Find each product or quotient. Write
your answers in simplest form.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
12(3x2 1 6x 2 4)
27(11x3 2 2x 1 18)
(12x2 1 4x 1 16) 4 4
24(22x4 2 3x3 1 4x2 1 2x 2 20)
(25x2 1 10x 2 5) 4 (25)
20(10x3 2 5x 1 11)
(35x2 1 28x 2 14) 4 7
(54x4 2 36x3 2 27x2 1 9x 2 18) 4 9
(62x2 1 2x 2 4) 4 (22)
28(5x4 1 16x2 1 8x 2 1)
88(10x2 2 9x)
(50x2 2 30x 1 40) 4 10
(121x2 1 11x) 4 11
Copyright © 2009 by Nelson Education Ltd.
Reproduction permitted for classrooms
Extending
8. Write each expression in simplest
form.
a)
b)
c)
d)
e)
f)
2(10x2 1 15) 4 10
25(12x4 1 7x3 1 28x2 1 24x) 4 5
7(x2 1 11 1 8x) 1 3(x2 2 4)
3(3x2 1 6x 2 9) 4 9
2(2(2(x 1 1))) 4 8
2(x 2 1 1) 2 3(x 1 1)
9. A rectangle has dimensions 3 m by
2x2 1 4x 1 8 metres. What is the area
of each of the triangles formed by
drawing one diagonal of the rectangle?
10. If c and d are whole numbers
and neither c nor d is 0, what
must be true about c and d if the
coefficients of c(2x2 1 3x 1 1) 4 d
are whole numbers?
8C Multiplying and Dividing an Algebraic Expression
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