Wize High School Algebra II Textbook (Common Core) > Rational Functions

Rational Expressions and Functions

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Rationa Expressions & Functions
A rational expression is a quotient (or ratio) of two polynomials, where the deonominator is not 0.

Examples
2x+1x−5,  x≠5\dfrac{2x+1}{x-5},~~x\neq5x−52x+1​,  x=5
3x2+2x−15x3−5,  x≠1\dfrac{3x^2+2x-1}{5x^3-5},~~x\neq15x3−53x2+2x−1​,  x=1
1x2+3\dfrac{1}{x^2+3}x2+31​


A rational function is any function that is a ratio of two polynomials, where the denominator is not 0.
f(x)=P(x)Q(x),  Q(x)≠0f(x)=\dfrac{P(x)}{Q(x)},~~Q(x)\neq 0f(x)=Q(x)P(x)​,  Q(x)=0
Examples
f(x)=2x+1x−5,  x≠5f(x)=\dfrac{2x+1}{x-5},~~x\neq5f(x)=x−52x+1​,  x=5
g(x)=3x2+2x−15x3−5,  x≠1g(x)=\dfrac{3x^2+2x-1}{5x^3-5},~~x\neq1g(x)=5x3−53x2+2x−1​,  x=1
h(x)=1x2+3h(x)=\dfrac{1}{x^2+3}h(x)=x2+31​

Simplifying Rational Expressions & Functions
We can simplify rational expressions and rational functions by dividing out any common factors that belong to both numerator and denominator.

Example
Simplify x2+3x+2x2−1\dfrac{x^2+3x+2}{x^2-1}x2−1x2+3x+2​

First, we factor fully:

=(x+1)(x+2)(x+1)(x−1)=\dfrac{(x+1)(x+2)}{(x+1)(x-1)}=(x+1)(x−1)(x+1)(x+2)​

Then, we divide out any common factor:

=(x+1)(x+2)(x+1)(x−1)=\dfrac{\cancel{(x+1)}(x+2)}{\cancel{(x+1)}(x-1)}=(x+1)​(x−1)(x+1)​(x+2)​

=x+2x−1=\dfrac{x+2}{x-1}=x−1x+2​

The restrictions are that

Practice: Evaluating Rational Functions
Consider f(x)=x2−1x−1f(x)=\dfrac{x^2-1}{x-1}f(x)=x−1x2−1​.
a) evaluate f(2)f(2)f(2)
b) evaluate f(−2)f(-2)f(−2)
c) state any restrictions on the variable xxx.

Evaluate f(2)f(2)f(2).

I don't know

Practice: Simplifying Rational Expressions
Consider the rational expression 4x2−12x+92x2−x−3\dfrac{4x^2-12x+9}{2x^2-x-3}2x2−x−34x2−12x+9​.

Simplfy this expression fully.

To enter a fraction

\dfrac{a}{b}

, enter a / b

I don't know

Practice: Simplifying Rational Expressions
Consider the rational expression x3−3x2y−10xy2x2−25y2\dfrac{x^3-3x^2y-10xy^2}{x^2-25y^2}x2−25y2x3−3x2y−10xy2​.

Simplify the expression fully.

To enter

\dfrac{a}{b}

, type a/ b

I don't know

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Practice: Simplifying Rational Expressions
Is there a rational expression that simplifies to x+1x−2\dfrac{x+1}{x-2}x−2x+1​, with denominator of the form ax2+bx+cax^2+bx+cax2+bx+c, and has the restriction

a) x≠−1x\neq -1x=−1 and x≠2x\neq 2x=2 only?

b) x≠2x\neq 2x=2 only?

I have answered this question