Bell Quiz 8-4
Factor the expression.
5 pts
1. 10x – 5x2
5 pts
2. x2 – 2x – 48
10 pts
possible
Section 8-5
1
8-5
Chapter 8: Rational Functions
Section 8-5
2
8-5
Chapter 8: Rational Functions
Section 8-5
3
Key Concept
Adding or Subtracting with Like Denominators
To add or subtract rational expressions with like denominators,
simply add (or subtract) their numerators and carry the
denominator across the equal sign.
Property:
ࢇ
ࢉ
࢈
ࢉ
൅ ൌ
ࢇା࢈
ࢉ
૜
૞
૚
૞
Example: ൅ ൌ
૜ା૚
૞
ൌ
૝
૞
__________________________________________________
Property:
ࢇ
ࢉ
࢈
ࢉ
െ ൌ
ࢇି࢈
ࢉ
Example:
Section 8-5
૜࢞
૞࢞૛
൅
ૠ
૞࢞૛
ൌ
૜࢞ାૠ
૞࢞૛
4
EXAMPLE 1
Add or subtract with like denominators
Perform the indicated operation.
a. 7 + 3
b. 2x – 5
x+6
x+6
4x
4x
Section 8-5
5
EXAMPLE 1
Add or subtract with like denominators
Perform the indicated operation.
a. 7 + 3
b. 2x – 5
x+6
x+6
4x
4x
SOLUTION
a. 7 + 3 = 7 + 3 = 10 = 5
4x
4x 2x
4x
4x
Add numerators and
simplify result.
b. 2x – 5 = 2x – 5
x+6
x+6 x+6
Subtract numerators.
Section 8-5
6
GUIDED PRACTICE
for Example 1
Perform the indicated operation and simplify.
a.
7–5 = 2 = 1
7 – 5
=
12x
12x 6x
12x 12x
Subtract numerators
and simplify results .
b.
2+1 = 3 = 1
2 + 1
=
3x2
3x2
x2
3x2 3x2
Add numerators and
simplify results.
c.
4x–x = 3x = 3x Subtract numerators.
4x – x
=
x–2
x–2 3x – 2
x–2 x–2
Factor numerators and
simplify results .
d. 24x + 22 = 4x2 + 2
x +1 x +1
x +1
Section 8-5
7
Key Concept
Adding or Subtracting with Unlike Denominators
To add (or subtract) two rational expressions with unlike
denominators, find a common denominator. Rewrite each rational
expression using the common denominator. Then add (or subtract).
You can always find a common denominator by multiplying the 2
denominators together.
Property
௔
௖
௕
ௗ
൅ ൌ
௔ௗ
௖ௗ
൅
௕௖
௖ௗ
ൌ
௔ௗା௕௖
௖ௗ
Section 8-5
௔
௖
௕
ௗ
െ ൌ
௔ௗ
௖ௗ
െ
௕௖
௖ௗ
ൌ
௔ௗି௕௖
௖ௗ
8
EXAMPLE 2
Find a least common multiple (LCM)
Find the least common multiple of 4x2 –16 and
6x2 –24x + 24.
Section 8-5
9
EXAMPLE 2
Find a least common multiple (LCM)
Find the least common multiple of 4x2 –16 and
6x2 –24x + 24.
SOLUTION
STEP 1
Factor each polynomial. Write numerical factors as
products of primes.
4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2)
6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2
Section 8-5
10
EXAMPLE 2
Find a least common multiple (LCM)
STEP 2
Form the LCM by writing each factor to the highest
power it occurs in either polynomial.
LCM = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2
Section 8-5
11
EXAMPLE 3
Add with unlike denominators
x
Add: 7 +
9x2 3x2 + 3x
Section 8-5
12
EXAMPLE 3
Add with unlike denominators
x
Add: 7 +
9x2 3x2 + 3x
SOLUTION
To find the LCD, factor each denominator and write
each factor to the highest power it occurs. Note that
9x2 = 32x2 and 3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1)
= 9x2(x 1 1).
7
7 +
x
x
Factor second denominator.
=
+
9x2
3x(x + 1)
9x2 3x2 + 3x
7
9x2
x+1
x
+
x + 1 3x(x + 1)
3x
3x
LCD is 9x2(x + 1).
Section 8-5
13
EXAMPLE 3
Add with unlike denominators
3x2
7x + 7
= 9x2(x + 1)+ 9x2(x + 1)
Multiply.
2 + 7x + 7
3x
=
9x2(x + 1)
Add numerators.
Section 8-5
14
EXAMPLE 4
Subtract:
Subtract with unlike denominators
x + 2 – –2x –1
2x – 2
x2 – 4x + 3
Section 8-5
15
EXAMPLE 4
Subtract with unlike denominators
x + 2 – –2x –1
2x – 2
x2 – 4x + 3
Subtract:
SOLUTION
x+2 –
2x – 2
x+2
= 2(x – 1)–
x+2
= 2(x – 1)
–2x –1
x2 – 4x + 3
– 2x – 1
(x – 1)(x – 3)
Factor denominators.
x – 3 – – 2x – 1
2
x – 3 (x – 1)(x – 3) 2
2–x–6
x
– 4x – 2
= 2(x – 1)(x – 3) –
2(x – 1)(x – 3)
Section 8-5
LCD is 2(x − 1)(x − 3).
Multiply.
16
EXAMPLE 4
Subtract with unlike denominators
2 – x – 6 – (– 4x – 2)
x
=
2(x – 1)(x – 3)
Subtract numerators.
2 + 3x – 4
x
=
2(x – 1)(x – 3)
Simplify numerator.
(x –1)(x + 4)
= 2(x – 1)(x – 3)
Factor numerator.
Divide out common
factor.
x+4
= 2(x
–3)
Simplify.
Section 8-5
17
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
5. 5x3 and 10x2–15x
Section 8-5
18
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
5. 5x3 and 10x2–15x
STEP 1
Factor each polynomial. Write numerical factors as
products of primes.
5x3 = 5(x) (x2)
10x2 – 15x = 5(x) (2x – 3)
STEP 2
Form the LCM by writing each factor to the highest
power it occurs in either polynomial.
LCM = 5x3 (2x – 3)
Section 8-5
19
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
6. 8x – 16 and 12x2 + 12x – 72
Section 8-5
20
GUIDED PRACTICE
for Examples 2, 3 and 4
Find the least common multiple of the polynomials.
6. 8x – 16 and 12x2 + 12x – 72
STEP 1
Factor each polynomial. Write numerical factors as
products of primes.
8x – 16 = 8(x – 2) = 23(x – 2)
12x2 + 12x – 72 = 12(x2 + x – 6) = 4 3(x – 2 )(x + 3)
STEP 2
Form the LCM by writing each factor to the highest
power it occurs in either polynomial.
LCM = 8 3(x – 2)(x + 3)
= 24(x – 2)(x + 3)
Section 8-5
21
GUIDED PRACTICE
7. 3 –
4x
8.
for Examples 2, 3 and 4
=
1
7
21 – 4x
28x
x
1 +
3x2 9x2 – 12x
x2 + 3x – 4
3x2 (3x – 4)
Section 8-5
22
GUIDED PRACTICE
9.
for Examples 2, 3 and 4
x
5
+
12x – 48
x2 – x – 12
=
Section 8-5
17x + 15
12(x +3)(x - 4)
23
HOMEWORK
Sec 8-5 (pg 586)
3-30 every 3rd, 41
Section 8-5
24