Rational Expressions and Equations

Reducing

Simplify the rational expression

\frac{2 x^{2} - 5 x - 3}{x^{2} - 4}

More examples

Cancelling the Common Factors

Simplify the rational expression

\frac{2 x^{2} y^{3}}{4 x^{3} y}

More examples

Rewriting in Standard Form

Rewrite the rational expression

\frac{3 x^{2} + 5 x - 2}{x^{2} - 4}

in standard form.

More examples

Operations on Rational Expressions

f (x) = \frac{x^{2} - 4 x + 4}{x - 2}

, simplify

f (x)

More examples

Determining if the Point is a Solution

Is the point (2,4) a solution to the equation

y = \frac{3 x - 2}{x + 1}

More examples

Finding the Domain

Find the domain of the function

f (x) = \frac{1}{x^{2} - 4}

More examples

Solving over the Interval

Solve the given rational equation for

x

over the interval

[0, 2 π]

\frac{1}{x} = \sin (x)

More examples

Finding the Range

Given the rational function

f (x) = \frac{(x - 1)}{(x + 2)}

, find the range of

f (x)

More examples

Finding the Domain and Range

Question 3 Let

f (x) = \frac{3 x - 6}{x^{2} + 8 x + 15} = \frac{(3 x - 6)}{(x + 5) (x + 3)}

Find: 1) the domain in interval notation Note: Use -oo for

- \infty

, oo for

\infty

, U for union. 2) the

y

intercept at the point 3)

x

intercepts at the point(s) 4) Vertical asymptotes at

x =

5) Horizontal asymptote at

y =

More examples

Solving Rational Equations

Solve the rational equation:

\frac{2 x}{x - 1} - \frac{3}{x} = 1

More examples

Adding Rational Expressions

\frac{4 x^{2} - 4 x - 9}{(2 x + 1) (x - 1)} \equiv A + \frac{B}{2 x + 1} + \frac{C}{x - 1}

a Find the values of the constants

A, B

and

C

. b Hence, or otherwise, expand

\frac{4 x^{2} - 4 x - 9}{(2 x + 1) (x - 1)}

in ascending powers of

x

, as far as the

x^{2}

term c Explain why the expansion is not valid for

x = \frac{3}{4}

More examples

Subtracting Rational Expressions

Subtract the rational expressions

\frac{5 x}{x + 2}

and

\frac{3 x}{x - 2}

More examples

Multiplying Rational Expressions

Multiply the following rational expressions:

\frac{4 x^{2} - 9}{3 x^{2} - 5 x + 2}

and

\frac{2 x - 1}{2 x^{2} + 5 x - 3}

More examples

Finding the Equation Given the Roots

Find the equation of a polynomial given the roots are

2

- 3

, and

1

More examples

Finding the Asymptotes

Let

f (x) = \frac{3 x - 6}{x^{2} + 8 x + 15} = \frac{(3 x - 6)}{(x + 5) (x + 3)}

Find: 3)

x

intercepts at the point(s) 4) Vertical asymptotes at

x = - 3

5) Horizontal asymptote at

y = - 5

More examples

Finding the Constant of Variation

If the variation equation is given as

y = k x^{2}

, and the point

(2, 8)

lies on the graph of this equation, what is the constant of variation

k

More examples

Finding the Equation of Variation

If y varies directly with x, and y = 8 when x = 2, find the equation of variation.

More examples

Reducing

Cancelling the Common Factors

Rewriting in Standard Form

Operations on Rational Expressions

Determining if the Point is a Solution

Finding the Domain

Solving over the Interval

Finding the Range

Finding the Domain and Range

Solving Rational Equations

Adding Rational Expressions

Subtracting Rational Expressions

Multiplying Rational Expressions

Finding the Equation Given the Roots

Finding the Asymptotes

Finding the Constant of Variation

Finding the Equation of Variation

Popular Problems