Rational Expressions and Equations

Rational equations and expressions pertain to the ratios of polynomial expressions. Essentially, these are algebraic fractions where both the top and bottom components are polynomials. The process of solving these equations usually involves identifying a common denominator, streamlining or minimizing the expression, and equating the numerator to zero in order to solve for any unknown variables.

Reducing

Simplify the expression \(\frac{9x^2 - 16}{3x-4}\)

Cancelling the Common Factors

Simplify the rational expression: \(\frac{9x^2 - 16}{3x + 4}\)

Rewriting in Standard Form

Rewrite the rational expression \(\frac{2x^2 - 5x - 3}{x^2 - 4}\) in standard form.

Operations on Rational Expressions

$\frac{x+\frac{1}{2}}{2+\frac{1}{x}}=$

Determining if the Point is a Solution

Is the point (3, -2) a solution to the equation \(y=\frac{2x-5}{x+1}\)?

Finding the Domain

RATIONAL EXPRESSIONS Restriction on a variable in a denominator: Quadratic Find all excluded values for the expression. That is, find all values of $x$ for which the expression is undefined. \[ \frac{x^{2}-6 x+8}{x^{2}-1} \] If there is more than one value, separate them with commas. \[ x= \]

Solving over the Interval

Solve the equation \(\frac{3}{x-2} - \frac{2}{x+3} = 1\) over the interval \([-5, 4]\)

Finding the Domain and Range

Find the domain and range of the function \(f(x) = \frac{1}{x+3}\)

Solving Rational Equations

$\frac{x}{(x+2)(x-2)}+\frac{4}{x-2}=\frac{3}{x+2}$

Adding Rational Expressions

$\frac{5}{a^{2}+12 a+36}+\frac{8}{a^{2}-36}$

Subtracting Rational Expressions

$\frac{6 n}{n^{2}-9}-\frac{3}{n+3}$

Multiplying Rational Expressions

Multiply the following rational expr \[ \frac{x^{2}+2 x-15}{x^{2}+(-4) x-12} \cdot \frac{x^{2}+8 x+12}{x^{2}-9} \]

Finding the Equation Given the Roots

Use the rational zeros theorem to list all possible zeros of the function $f(x)=7 x^{3}-3 x^{2}-x+2$ Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Finding the Asymptotes

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

Finding the Constant of Variation

If \(y = kx^2\), where \(k\) is a constant, and \(y = 25\) when \(x = 5\), what is the value of \(k\)?

Finding the Equation of Variation

If y varies directly as the square of x and inversely as the cube root of z, where y = 9 when x = 3 and z = 8, find the equation of variation.