Embed Size (px)
Citation preview
UNIT: RATIONAL FUNCTIONS9-4: RATIONAL EXPRESSIONSDAY 1 (SIMPLIFICATION)Essential Question: How do you simplify, multiply and divide rational expressions?
9-4: Rational Expressions
A rational expression is in simplest form when its numerator and denominator are polynomials which have no common divisors.
Examples in simplest form:
Not in simplest form:
2
3
1 3
x
x x
1
2
2( 3)
1 3( 3)xx x
x x x
9-4: Rational Expressions
To simplify a rational expression: Remove any GCFs that may exist Factor if possible Cancel common factors (parenthesis must
match) To find restrictions:
Before canceling out anything, check the denominators
Set any denominator pieces equal to 0 These give you numbers your variable cannot
be
9-4: Rational Expressions
Example Simplify . State any restrictions
on the variables. Factor x2 + 10x + 25
Factor x2 + 9x + 20
Restrictions:
2
2
10 25
9 20
x x
x x
(x + 5)(x + 5)
(x + 4)(x + 5)( 5)( 5)
( 4)( 5)
x x
x x
( 5)
( 4)
x
x
x ≠ -4, x ≠ -5
9-4: Rational Expressions
Your Turn #1 (Already simplified, just reduce) Simplify . State any restrictions on
the variables.
Restrictions: Simplified:
3
4
27
9
x y
x y
x ≠ 0, y ≠ 03
x
9-4: Rational Expressions
Your Turn #2 (GCF & Factor) Simplify . State any restrictions
on the variables. Simplify -6 – 3x
Factor x2 – 6x + 8
Restrictions:
2
6 3
6 8
x
x x
x ≠ 2, x ≠ 4
-3(x + 2)
(x – 2)(x – 4)3( 2)
( 2)( 4)
x
x x
9-4: Rational Expressions
Your Turn #3 (Factor top & bottom) Simplify . State any restrictions
on the variables. Factor 2x2 – 3x – 2
Factor x2 – 5x + 6
Restrictions:
2
2
2 3 2
5 6
x x
x x
(2x + 1)(x – 2)
(x – 2)(x – 3)(2 1)( 2)
( 2)( 3)
x x
x x
(2 1)
( 3)
x
x
x ≠ 2, x ≠ 3
9-4: Rational Expressions
Assignment Page 511 Problems 1 – 6, all You must show your work
UNIT: RATIONAL FUNCTIONS9-4: RATIONAL EXPRESSIONSDAY 2 (MULTIPLICATION)Essential Question: How do you simplify, multiply and divide rational expressions?
9-4: Rational Expressions
To multiply rational expressions: Remove any GCFs that may exist Factor if possible Cancel common factors (parenthesis must match) Multiply numerators with numerators,
denominators with denominators To find restrictions:
Before canceling out anything, check all denominators
Set any denominator pieces equal to 0 These give you numbers your variable cannot be
9-4: Rational Expressions
Example Multiply and . State
any restrictions on the variables. Factor 2x2 + 7x + 3
Factor x2 – 16
Factor x2 + 8x + 15
Simplified:
Restrictions:
22 7 3
4
x x
x
2
2
16
8 15
x
x x
(2x + 1)(x + 3)
(x – 4)(x + 4)
x ≠ 4, x ≠ -5, x ≠ -3
(2 1)( 3) ( 4)( 4)
4 ( 5)( 3)
x x x x
x x x
(x + 5)(x + 3)(2 1)( 4)
( 5)
x x
x
9-4: Rational Expressions
YOUR TURN Multiply and . State any restrictions on
the variables.
Factor x2 – 4
Factor x2 – 1
Factor x2 + 2x
Simplified:
Restrictions:
2
2
4
1
x
x
2
1
2
x
x x
(x + 2)(x – 2)
(x + 1)(x – 1)
x ≠ -1, x ≠ 1, x ≠ 0, x ≠ -2
( 2)( 2) ( 1)
( 1)( 1) ( 2)
x x x
x x x x
x(x + 2)( 2)
( 1)
x
x x
UNIT: RATIONAL FUNCTIONS9-4: RATIONAL EXPRESSIONSDAY 3 (DIVISION)Essential Question: How do you simplify, multiply and divide rational expressions?
9-4: Rational Expressions
To divide rational expressions: Remove any GCFs that may exist Factor if possible Flip division sign to multiplication, flip fraction after
division sign Cancel common factors (parenthesis must match) Multiply numerators with numerators,
denominators with denominators To find restrictions:
Before canceling out anything, check all denominators
Set any denominator pieces equal to 0 These give you numbers your variable cannot be Restrictions must be checked both before and
after the flip
9-4: Rational Expressions
Example Divide by . State
any restrictions on the variables. Fix 4 – x (always want x to come first and be
positive)
Flip division into multiplication
Simplified:
Restrictions:
4
(3 2)( 2)
x
x x
5( 4)
( 2)(7 5)
x
x y
-1(x – 4)
x ≠ -2/3, x ≠ 2, y ≠ 5/7
1( 4) 5( 4)
(3 2)( 2) ( 2)(7 5)
x x
x x x y
(7 5)
5(3 2)
y
x
1( 4) ( 2)(7 5)
(3 2)( 2) 5( 4)
x x y
x x x
, x ≠ 4
9-4: Rational Expressions
YOUR TURN Divide by . State any restrictions on the
variables.
Factor x2 +2x – 15
Factor x2 – 16
Factor 3x – 12
Flip division into multiplication:
Restrictions: Simplified:
2
2
2 15
16
x x
x
1
3 12
x
x
(x + 5)(x – 3)
(x + 4)(x – 4)
( 5)( 3) ( 1)
( 4)( 4) 3( 4)
x x x
x x x
3(x – 4)
( 5)( 3) 3( 4)
( 4)( 4) ( 1)
x x x
x x x
x ≠ 4, x ≠ -4
, x ≠ -13( 5)( 3)
( 4)( 1)
x x
x x
9-4: Rational Expressions
Assignment Page 511 Problems 7 – 18, all You must show your work
