3.9 multiplying & dividing rational expressions p. 165-168
TRANSCRIPT
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