rational algebraic expressions objective: to simplify rational algebraic expressions
TRANSCRIPT
Rational Algebraic Expressions
Objective: To simplify rational algebraic expressions
Rational Algebraic Expression
• Quotient of two algebraic expressions
• Example: 2
2
2 1
1
x x
x
Simplify the following expression:
1. start by factoring
2. after the polynomials are factored you can
“cancel” terms whose quotient is one
3.
2
2
2 1
1
x x
x
( 1)( 1)
( 1)( 1)
x x
x x
( 1)
( 1)
x
x
Finding domain and zero’s
• Given:
• Start by factoring the numerator and denominator.
• The domain of the function is all reals except where the denominator equals zerox ≠ 0 x + 2 ≠ 0 x – 2 ≠ 0x ≠ 0 x ≠ -2 x ≠ 2
2
3
2 3 2( )
4
x xf x
x x
(2 1)( 2)( )
( 2)( 2)
x xf x
x x x
Zeros of the function
• The zeros of the function are where the numerator equals zero.
• 2x + 1 = 0 x – 2 = 0x = -½ x = 2
• Remember x = 2 is not in the domain so it cannot be a zero of the function.
Try These!Simplify
1.
2.
3.
4.
5.
1.
2. 2x – 1
3.
4.
5.
2
2
25
5
x
x x
5x
x
24 1
2 1
x
x
12 25 14 4 77x x x x
2
11
x
x
4 4
3 2 2 3
x y
x x y xy y
2 2x y
x y
4 2
3
1
1
x x
x
#5 Hint: add and subtract all missing terms
2 1
1
x x
x
More try these!
2
2
25
5
x
x x
( 5)( 5)
( 5)
x x
x x
( 5)x
x
Back to Try These!
24 1
2 1
x
x
(2 1)(2 1)
2 1
x x
x
2 1x
Back to Try These!
12 25 14 4 77x x x x
2
2
5 14
4 77
x x
x x
( 7)( 2)
( 11)( 7)
x x
x x
( 2)
( 11)
x
x
Back to Try These!
4 4
3 2 2 3
x y
x x y xy y
2 2( )( )( )
( )( )( )
x y x y x y
x y x y x y
2 2x y
x y
Back to Try These!
4 2
3
1
1
x x
x
4 3 3 2 2 2
3
1
1
x x x x x x x x
x
4 3 2 3 2 2
3
1
1
x x x x x x x x
x
2 2 2 2
3
( 1) ( 1) 1( 1)
1
x x x x x x x x
x
2 2
2
( 1)( 1)
( 1)( 1)
x x x x
x x x
2( 1)
( 1)
x x
x
Back to Try These!
Try These!Find the domain and zeros of the
function1.
2.
3.
1. D: x ≠ 0, x ≠ -4zeros {2, -2}
2. D: x ≠ -1/3 x ≠ -4zeros {-3/2, -2}
3. D: x ≠ -1zeros {-2}
2
2
4( )
4
xf x
x x
2
2
3 2( )
2 1
x xf x
x x
2
2
2 7 6
3 13 4
x x
x x