3.9 Multiplying & Dividing Rational Expressions p. 165-168.
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Transcript of 3.9 Multiplying & Dividing Rational Expressions p. 165-168.
Multiplying Factor numerator and denominator of each
expression Combine both numerators on top Combine both denominators on bottom Look to cancel any common factors Simplify
Multiply rational expressions involving monomialsEXAMPLE 1
Find the product 2x2
3x
6x2
12x3
Multiply numerators and
denominators.
Product of powers property
Factor and divide out common factors.
Simplify.13=
12 x4
3 12 x4 =
12x4
36x4=
2x2
3x
6x2
12x3=
(2x2)(6x2)
(3x)(12x3)
GUIDED PRACTICE for Example 1
Find the product
2y3
5y
15y3
8y5
1.
34ANSWER
2.7z2
4z3
z3
14z
ANSWERz8
EXAMPLE 2 Multiply rational expressions involving polynomials
Find the product 3x2 + 3x4x2 – 24x + 36
x2 – 4x + 3x2 – x
Multiply numerators and denominators.
3x2 + 3x4x2 – 24x + 36
x2 – 4x + 3x2 – x
=(3x2 + 3x) (x2 – 4x + 3)(4x2 – 24x + 36)(x2 – x)
Factor and divide out common factors.
= 3(x + 1)4(x – 3)
Simplify.
= 3x(x + 1)(x – 3)(x – 1)4x(x – 3)(x – 3)(x – 1)
EXAMPLE 2 Multiply rational expressions involving polynomials
CHECK
Check your simplification using a graphing calculator.
The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3).
Graph y1 = 3x2 + 3x4x2 – 24x + 36
x2 – 4x + 3x2 – x
and y2 =3(x + 1)4(x – 3)
.
EXAMPLE 3 Multiply a rational expression by a polynomial
Find the product 5x
x2 + 5x + 6(x + 3).
5x
x2 + 5x + 6(x + 3).
5x
x2 + 5x + 6
(x + 3)=
1
Rewrite polynomial as a fraction.
= 5x(x + 3)x2 + 5x + 6
Multiply numerators and denominators.
=5x(x + 3)
(x + 3)(x + 2)
Factor and divide out common factor.
Simplify.=5x
x + 2
GUIDED PRACTICE for Examples 2 and 3
3. Find the product x2 + x – 2 x2 + x
2x2 + 2x5x2 –15x +10
2(x + 2)5(x – 2)
ANSWER
Find the product2w2
w2 – 7w + 12(w – 4).4.
2w2
w – 3ANSWER
Dividing Multiply by the multiplicative inverse
(reciprocal) of the second expression Factor & write as one rational expression Cancel/divide out common factors Simplify
EXAMPLE 4 Divide rational expressions involving polynomials
Find the quotient .7x2 – 7xx2 + 2x – 3
x + 1x2 – 7x – 8
7x2 – 7xx2 + 2x – 3
x + 1x2 – 7x – 8
7x2 – 7xx2 + 2x – 3 x + 1
= x2 – 7x – 8 Multiply by multiplicative inverse.
= (7x2 – 7x) (x2 – 7x – 8)(x + 1)(x2 + 2x – 3)
Multiply numerators and denominators.
= 7x(x – 1)(x – 8)(x + 1)(x + 3)(x – 1)(x + 1)
Factor and divide out common factors.
=7x(x – 8)
x + 3Simplify.
EXAMPLE 5 Divide a rational expression by a polynomial
Find the quotient 2x2 + 16x + 243x2
(x +6).
2x2 + 16x + 243x2
(x +6)=
2x2 + 16x + 243x2
(x +6).= 1
Rewrite polynomial as fraction.
2x2 + 16x + 243x2=
1(x +6).
Multiply by multiplicative inverse.
2x2 + 16x + 243x2= (x +6).
Multiply numerators and denominators.
=2(x + 2)(x + 6)
3x2(x + 6)Factor and divide out common factor.
=2(x + 2)
3x2 Simplify.
GUIDED PRACTICE for Examples 4 and 5
5. m2 – 4
2m2 + 4m
6m – 3m2
4m + 44
2 (m +11)
3m2
–ANSWER
6. n2 – 6n + 912n
n – 3
ANSWER12n
n – 3
Find the quotient.
EXAMPLE 6 Solve a multi-step problem
A = 13,000 + 3700x 1 – 0.015x
and T = 1800 + 860x1 – 0.016x
The amount A (in millions of dollars) spent on all advertising and the amount T (in millions of dollars) spent on television advertising in the United States during the period 1970–2003 can be modeled by
where x is the number of years since 1970. Write a model that gives the percent p (in decimal form) of the amount spent on all advertising that was spent on television advertising. Then approximate the percent spent on television advertising in 2003.
ADVERTISING
EXAMPLE 6 Solve a multi-step problem
STEP 1
SOLUTION
Write a verbal model. Then write an equation.
P = T A
p = T A Write equation.
1800 +860x1– 0.016
13,000 + 3700x1– 0.015x x
=
STEP 2 Find the quotient.
Substitute for T and for A.
EXAMPLE 6 Solve a multi-step problem
1800 +860x1–0.016x 13,000 3700x+
1– 0.015x=
(1800 + 860x)(1 –0.015x)
(1– 0.016x)(13,000 + 3700x)=
Multiply by multiplicative inverse.
Multiply numerators and denominators.
Factor and divide out common factor.
(90 + 43x)(1 – 0.015x)
(1 – 0.016x)(650 + 185x)= Simplify.
20 (90 + 43x) (1 – 0.015x)
(1 – 0.016x)(20)(650 + 185x)=
EXAMPLE 6 Solve a multi-step problem
STEP 3 Approximate the percent spent on television advertising in 2003.Because 2003 – 1970 = 33, x = 33. Substitute 33 for x in the model and use a calculator to evaluate.
ANSWER About 24% of the amount spent on all advertising was spent on television advertising in 2003.
(90 + 43 33)(1 – 0.015 33)
(1 – 0.016 33)(650 + 185 33)P = 0.239