Download § 6.1 Rational Functions and Simplifying Rational Expressions

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§ 6.1 RATIONAL FUNCTIONS
AND SIMPLIFYING RATIONAL
EXPRESSIONS
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Rational Expressions
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Rational Expressions
Fractions that are the quotient of two
polynomials, where the denominator is not
equal to 0, are called rational expressions.
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Rational Expressions
Fractions that are the quotient of two
polynomials, where the denominator is not
equal to 0, are called rational expressions.
E.g.:
7𝑥 2 +11𝑥+8
𝑥−2
𝑥 2 −3
𝑥 4 +8𝑥 3 −17
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Rational Expressions
Fractions that are the quotient of two
polynomials, where the denominator is not
equal to 0, are called rational expressions.
E.g.:
7𝑥 2 +11𝑥+8
𝑥−2
𝑥 2 −3
𝑥 4 +8𝑥 3 −17
3𝑥 + 13 3
13
= 𝑥+
4
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Domain of a Rational Function
Definition of a Rational Function: A function whose
equation is defined by a rational expression in one
variable, where the value of the polynomial in the
denominator is never zero.
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Domain of a Rational Function
Definition of a Rational Function: A function whose
equation is defined by a rational expression in one
variable, where the value of the polynomial in the
denominator is never zero.
The domain of a rational function is all real
numbers except the real numbers that make the
polynomial in the denominator equal to 0.
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Finding the Domain of a Rational
Function
1.) Set the denominator equal to zero.
2.) Solve, factor if necessary.
3.) If the polynomial is not solvable, then the
domain is all real numbers.
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Example: Finding the Domain
𝑥 2 + 3𝑥 + 2
𝑥 2 − 𝑥 − 56
1.) 𝑥 2 − 𝑥 − 56 = 0
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Example: Finding the Domain
𝑥 2 + 3𝑥 + 2
𝑥 2 − 𝑥 − 56
1.) 𝑥 2 − 𝑥 − 56 = 0
2.) 𝑥 + 7 𝑥 − 8 = 0
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Example: Finding the Domain
𝑥 2 + 3𝑥 + 2
𝑥 2 − 𝑥 − 56
1.) 𝑥 2 − 𝑥 − 56 = 0
2.) 𝑥 + 7 𝑥 − 8 = 0
𝑥+7 =0
𝑥 = −7
𝑥−8 =0
𝑥=8
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Example: Finding the Domain
𝑥 2 + 3𝑥 + 2
𝑥 2 − 𝑥 − 56
1.) 𝑥 2 − 𝑥 − 56 = 0
2.) 𝑥 + 7 𝑥 − 8 = 0
𝑥+7 =0
𝑥−8 =0
𝑥 = −7
𝑥=8
Answer: All real number except -7,8.
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Example: Finding the Domain
𝑥 2 + 3𝑥 + 2
𝑥 2 − 𝑥 − 56
1.) 𝑥 2 − 𝑥 − 56 = 0
2.) 𝑥 + 7 𝑥 − 8 = 0
𝑥+7 =0
𝑥−8 =0
𝑥 = −7
𝑥=8
Answer: All real number except -7,8.
−∞, −7 ∪ −7,8 ∪ 8, ∞
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Examples on Simplifying
1.)
49𝑥𝑦 2
21𝑥𝑦
2.)
18𝑚4
36𝑚4 −9𝑚3
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General Guidelines for Simplifying
Rational Expressions
1.) Factor out the G.C.F. from the numerator and
denominator.
2.) Factor the numerator and denominator
completely.
3.) Cancel products.
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General Guidelines for Simplifying
Rational Expressions
1.) Factor out the G.C.F. from the numerator and
denominator.
2.) Factor the numerator and denominator
completely.
3.) Cancel products.
1.)
𝑥 2 +6𝑥+9
2𝑥 2 +6𝑥
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General Guidelines for Simplifying
Rational Expressions
1.) Factor out the G.C.F. from the numerator and
denominator.
2.) Factor the numerator and denominator
completely.
3.) Cancel products.
1.)
𝑥 2 +6𝑥+9
2𝑥 2 +6𝑥
2.)
4𝑥 2 +24𝑥+32
16𝑥 2 +8𝑥−48
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General Guidelines for Simplifying
Rational Expressions
1.) Factor out the G.C.F. from the numerator and
denominator.
2.) Factor the numerator and denominator
completely.
3.) Cancel products.
1.)
𝑥 2 +6𝑥+9
2𝑥 2 +6𝑥
2.)
4𝑥 2 +24𝑥+32
16𝑥 2 +8𝑥−48
Recall: 𝑎 − 𝑏 = − −𝑎 + 𝑏 = −(𝑏 − 𝑎)
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Note about Simplifying
Remember: You can only cancel products not
terms!
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Note about Simplifying
Remember: You can only cancel products not
terms!
E.g.:
𝑥
𝑥+𝑦
≠
1
1+𝑦
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Note about Simplifying
Remember: You can only cancel products not
terms!
E.g.:
𝑥
𝑥+𝑦
≠
1
1+𝑦
𝑥
1
=
𝑥 𝑥+𝑦
𝑥+𝑦