Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
C. Operations on Rational
Expressions (pp. 109–114)
In Chapter 9 you learned how to add, subtract and multiply polynomials. You will now
learn how to divide polynomials and determine excluded values. The following examples
also show how to apply the operations of addition, subtraction, multiplication, and division
to rational expressions.
1. Divide a Polynomial by a Monomial
EXAMPLE
Divide 3y 3 1 9y 2 2 6y by 3y .
Method 1: Write the division as a fraction.
3y3 1 9y2 2 6y
(3y3 1 9y2 2 6y) 4 3y 5 }}
3y
3y3
9y2
Write as fraction.
6y
5}
1}
2}
3y
3y
3y
Divide each term by 3y.
5 y2 1 3y 2 2
Method 2: Use long division.
Think:
9y 4 3y 5 ?
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Think:
3
3y 4 3y 5 ?
Check your
answer by
multiplying:
3y(y 2 1 3y 2 2).
PRACTICE
Simplify.
Think:
26y 4 3y 5 ?
2
BENCHMARK 6
C. Rational Expressions
Solution:
y2 1 3y 2 2
3yqww
3y3 1 9y2 2 6y
(3y3 1 9y2 2 6y) 4 3y 5 y2 1 3y 2 2
Divide
1. (8x3 2 12x2 1 4x) 4 4x
2. (15y4 1 5y3 2 20y2) 4 5y
3. (10y3 1 6y2 2 20y) 4 2y
4. (12z4 2 3z3 2 15z2) 4 3z
2. Divide a Polynomial by a Binomial
EXAMPLE
Divide x 2 1 3x 2 10 by x 2 2.
Solution:
Be sure to subtract
x 2 2 2x from
x2 1 3x to obtain
5x.
Step 1: Divide the first term of x2 1 3x 2 10 by the first term of x 2 2.
x
x2 1 3x 2 10
Think: x2 4 x 5 ?
x 2 2 qww
x2 2 2x
5x
Multiply x and x 2 2.
Subtract x2 2 2x from x2 1 3x.
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
109
Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
Step 2: Bring down 210. Then divide the first term of 5x 2 10 by the first term of
x 2 2.
x15
x 2 2 qww
x2 1 3x 2 10
x2 2 2x
5x 2 10
5x 2 10
0
Think: 5x 4 x 5 ?
Multiply 5 and x 2 2.
Subtract 5x 2 10 from 5x 2 10.
EXAMPLE
Divide 2x2 1 7x 2 1 by 2x 2 1.
When you obtain a Solution:
nonzero remainder,
x14
apply the rule:
2x 2 1 qww
2x2 1 7x 2 1
Dividend 4 Divisor
5 Quotient 1
Multiply x and 2x 2 1.
2x2 2 x
Remainder
}.
8x 2 1
Subtract 2x2 2 x. Bring down 21.
Divisor
To check your
8x 2 4
Multiply 4 and 2x 2 1.
answer, multiply
the quotient by the
3
Subtract 8x 2 4 from 8x 2 1.
divisor, then add
3
the remainder to
(2x2 1 7x 2 1) 4 (2x 2 1) 5 x 1 4 1 }
2x 2 1
the product.
PRACTICE
Divide
5. (y2 1 y 2 12) 4 (y 1 4)
6. (b2 2 2b 2 8) 4 (b 1 2)
7. (3a2 1 8a 1 4) 4 (3a 2 1)
8. (2x2 2 5x 2 14) 4 (x 2 4)
3. Find the Excluded Values of Rational Expressions
Vocabulary
EXAMPLE
A rational
expression is
undefined if its
denominator is 0.
Rational expression An expression that can be written as a ratio of two polynomials
where the denominator is not 0.
Excluded value a number that makes a rational expression undefined.
Find the excluded values, if any, of the expression.
y25
a. }
4y
3
b. }
3b 2 15
2x
c. }
x2 2 4
3m 2 1
d. }}
5m2 1 2m 1 4
Solution:
y25
a. The expression } is undefined when 4y 5 0, or y 5 0.
4y
The excluded value is 0.
3
b. The expression } is undefined when 3b 2 15 5 0, or b 5 5.
3b 2 15
The excluded value is 5.
2x
c. The expression }
is undefined when x2 2 4 5 0, or (x 2 2) (x 1 2) 5 0.
x2 2 4
The solutions of the equation are 22 and 2.
The excluded values are 22 and 2.
110
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 6
C. Rational Expressions
(x2 1 3x 2 10) 4 (x 2 2) 5 x 1 5
Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
If the discriminant
of the quadratic
equation is
nonnegative,
then you must
solve it to find the
excluded values.
PRACTICE
3m 2 1
d. The expression }}
is undefined when 5m2 1 2m 1 4 5 0.
5m2 1 2m 1 4
The discriminant is b2 2 4ac 5 22 2 4(5)(4) 0. So, the quadratic equation has
no real roots.
There are no excluded values.
7x
12. }
8 2 2x
n13
16. }
n2 2 6n 1 9
4. Multiply and Divide Rational Expressions
Vocabulary
EXAMPLE
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Use a graphing
calculator to
check your
simplification.
Graph
x2 2 4x 1 4
+
y1 5 }
2
Rational expression An expression that can be written as a ratio of two polynomials
where the denominator is not 0.
Excluded value A number that makes a rational expression undefined.
x 2 2 4x 1 4
Find the product }}
+
2
x 1 3x
x2 2 4x 1 4
x 1 3x
2x2 1 2x
3x 2 3x 2 6
+}
}
2
2
(x2 2 4x 1 4)(2x2 1 2x)
(x 1 3x)(3x 2 3x 2 6)
Multiply numerators and denominators.
2x(x 2 2)(x 2 2)(x 1 1)
3x(x 1 3)(x 1 1)(x 2 2)
Factor and divide out common factors.
2(x 2 2)
3(x 1 3)
Simplify.
5 }}
2
2
x 1 3x
2x 1 2x
and
}
3x2 2 3x 2 6
2(x 2 2)
y2 5 }. If the
3(x 1 3)
5 }}
graphs coincide,
your simplification
is correct.
To divide by
a rational
expression,
multiply by its
multiplicative
inverse.
2
2
Solution:
2
EXAMPLE
2x 1 2x
.
}}
3x 2 3x 2 6
5}
Find the quotient
x2 2 5x 1 4
BENCHMARK 6
C. Rational Expressions
Find the excluded values, if any, of the expression.
2b 1 3
x
22
9. }
10. }
11. }
5y 1 15
5b
x2 2 25
z25
m2 2 1
4a
13. }
14. }
15. }
2m
z2 2 3z 2 4
a2 1 9
x 1x22
4}
.
}}
5x 1 20
x 1 4x
2
2
Solution:
x2 2 5x 1 4
x 1 4x
x2 1 x 2 2
5x 1 20
4}
}
2
5x 1 20
x 1x22
x 2 5x 1 4 }
+ 2
5}
2
2
x 1 4x
Multiply by multiplicative inverse.
(x2 2 5x 1 4)(5x 1 20)
(x 1 4x)(x 1 x 2 2)
Multiply numerators and denominators.
5(x 2 4)(x 2 1)(x 1 4)
x(x 1 4)(x 1 2)(x 2 1)
Factor and divide out common factors.
5(x 2 4)
x(x 1 2)
Simplify.
5 }}
2
2
5 }}
5}
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
111
Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
PRACTICE
Find the product or quotient.
x2 1 x 2 6
4x2 2 2x
17. }
+}
2
2
12x 2 6x x 2 x 2 12
z2 2 25
4z2 1 8z
18. } + }
2
2z 1 4 z 2 3z 2 10
x2 2 6x
x2 2 36
19. } 4 }
4x 1 12
x2 1 2x 2 3
w2 2 3w 2 4
w2 2 w 2 12
20. }
4}
2w2 1 4w
w3 1 2w2
5. Find the LCD of Rational Expressions
BENCHMARK 6
C. Rational Expressions
Vocabulary
EXAMPLE
Least common denominator (LCD) The LCD of two or more rational expressions
is the product of the factors of the denominators of the rational expressions with each
common factor used only once.
Find the LCD of the rational expressions.
3 22z
a. }2 , }
9z
6z
Be sure to use the
common factors
only once when
finding the LCD.
b.
7
a
a 2 4 a 2 4a 1 4
x23
2
c. }
,}
x2 1 4 3x 2 1
}
,}
2
2
Solution:
a. Find the least common multiple (LCM) of 6z2 and 9z.
6z2 5 2 + 3 + z + z
The common factors are circled.
9z 5 3 + 3 + z
LCM 5 2 + z + 3 + 3 + z 5 18z2
3
6z
22z
b. Find the least common multiple (LCM) of a2 2 4 and a2 2 4a 1 4.
You can use the
discriminant
of a quadratic
expression
to check for
factorability. If it
is negative, the
expression cannot
be factored.
PRACTICE
a2 2 4 5 (a 2 2) + (a 1 2)
a2 2 4a 1 4 5 (a 2 2) + (a 2 2)
LCM 5 (a 2 2) + (a 1 2) + (a 2 2) 5 (a 2 2)2 (a 1 2)
a
a 24
c. Find the least common multiple (LCM) of x2 1 4 and 3x 2 1.
Because x2 1 4 and 3x 2 1 cannot be factored, they don’t have any factors in
common. The least common multiple is their product, (x2 1 4)(3x 2 1).
x23
2
The LCD of }
and }
and is (x2 1 4)(3x 2 1).
2
3x 2 1
x 14
Find the LCD of the rational expressions.
x15 4
21. } ,}3
12x 18x
112
7
a 2 4a 1 4
The LCD of }
and }
is (a 2 2)2 (a 1 2).
2
2
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
22.
3b
2
b 12 b22
}
,}
2
32y
6
23. }
,}
2
y 1 2y 2 3 5y2 2 5
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
The LCD of }2 and }
is 18z2.
9z
Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
6. Add and Subtract Rational Expressions
EXAMPLE
Find the sum or difference.
7
2
a. } 1 }2
12y
16y
x11
x21
b. }
2}
x2 1 x 2 6
x2 2 4
Solution:
2 + 4y
7
7+3
2
a. } 1 }2 5 } 1 }
12y
12y + 4y
16y
16y2 + 3
21
5 }2 1 }2
Simplify numerators and
denominators.
48y
8y 1 21
48y
5}
2
Add fractions.
x11
x21
b. }
2}
x2 1 x 2 6
x2 2 4
x11
(x 2 2)(x 1 3)
x21
(x 2 2)(x 1 2)
5 }} 2 }}
(x 1 1)(x 1 2)
(x 2 2)(x 1 3)(x 1 2)
Factor denominators.
(x 2 1)(x 1 3)
(x 1 2)(x 2 2)(x 1 3)
5 }} 2 }}
(x 1 1)(x 1 2) 2 (x 2 1)(x 1 3)
(x 2 2)(x 1 3)(x 1 2)
Subtract fractions.
x2 1 3x 1 2 2 (x2 1 2x 2 3)
(x 2 2)(x 1 3)(x 1 2)
Find products in numerator.
x15
(x 2 2)(x 1 3)(x 1 2)
Simplify.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5 }}}
To subtract
x 2 1 2x 2 3 in the
numerator, add
the opposite of
every term.
PRACTICE
Rewrite fraction using LCD,
(x 2 2)(x 1 3)(x 1 2).
5 }}}
5 }}
Find the sum or difference.
3
4
24. }2 1 }
10b
4b
z11
z21
26. }
2}
z2 2 2z 2 15
z2 1 z 2 6
BENCHMARK 6
C. Rational Expressions
8y
48y
Rewrite fraction using LCD, 48y2.
5x
9
25. } 2 }
x23
4x
Quiz
Divide
1. (3a3 1 18a2 2 9a) 4 3a
2. (16x4 1 24x3 1 8x2) 4 4x
3. (20m3 2 5m2 2 15m) 4 5m
4. (14y4 2 6y3 1 18y) 4 2y
5. (n2 1 4n 2 12) 4 (n 2 2)
6. (2x2 2 5x 2 3) 4 (x 2 3)
7. (4b2 1 4b 2 7) 4 (2b 1 3)
8. (4y2 2 19y 1 1) 4 (y 2 5)
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
113
Name ———————————————————————
Date ————————————
BENCHMARK 6
(Chapters 11, 12, and 13)
Find the excluded values, if any, of the expression.
z12
9. }
7z
4a 1 3
11. }
a2 2 1
m2
10. }
2
m 1 16
3
12. }
3x2 1 x 1 6
a2 2 4
a2 2 2a
13. }
4}
3
a 2 16a
a3 2 4a2
y2 2 9
y2 2 1
14. }
+}
2
2
y 1 y y 1 5y 1 6
b3 2 4b2 2 5b
4b 1 12
15. }
+ }}
2
22b 2 6
2b 2 2b 2 4
3x2 2 12x 1 12
6x2 2 6x 2 12
16. }}
4 }}
3
2
x 2 4x
x2 2 4x
Find the LCD of the rational expressions.
2y 2 1 1
17. }
,}
15y3 25y2
18.
3
4m 2 1
2m 2 8 m 1 3m 2 10
}
, }}
2
2
7
x14
19. }
,}
x2 1 3x 1 3 3x
Find the sum or difference.
5
3
20. } 2 }3
12x
20x
21.
m
m12
7
3m
} 1 }2
b13
b12
22. }
2}
b2 2 16
b2 1 2b 2 8
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
BENCHMARK 6
C. Rational Expressions
Find the product or quotient.
114
Algebra 1
Benchmark 6 Chapters 11, 12, and 13
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