Easily identifies real zeros (x-intercepts), domain, vertical asymptotes, holes, and provides insight into the range.

Even degree with positive leading coefficient: ends go up. Even degree with negative leading coefficient: ends go down. Odd degree with positive leading coefficient: starts down, ends up. Odd degree with negative leading coefficient: starts up, ends down.

Compare degrees of numerator and denominator to find horizontal asymptotes, or the possibility of a slant asymptote.

To divide one polynomial by another, which helps in finding slant asymptotes and rewriting rational functions.

Pascal's Triangle provides the coefficients for the terms in the binomial expansion.

The remainder helps in expressing the original rational function as a sum of a polynomial and a simpler rational function, useful for identifying asymptotes.

Maxima, minima, points of inflection, shape, symmetry, asymptotes, and intercepts.

Holes occur when a factor cancels out from both the numerator and denominator.

Perform polynomial long division. If the degree of the numerator is one greater than the degree of the denominator, the quotient is the equation of the slant asymptote.

Zeros, y-intercept, vertical asymptotes, horizontal or slant asymptotes, and end behavior.

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

$f(x) = g(x)q(x) + r(x)$ where $q(x)$ is the quotient and $r(x)$ is the remainder.

$f(x) = q(x) + \frac{r(x)}{g(x)}$, where $q(x)$ represents the slant asymptote if the degree of $g(x)$ is 1.

Numerator degree < denominator degree.

Numerator degree = denominator degree.

Numerator degree > denominator degree.

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Set each factor equal to zero and solve for x.

Identify the leading term (term with the highest degree). Consider the sign and the exponent of the variable to determine the end behavior.

Divide the highest degree term of the dividend by the highest degree term of the divisor. Multiply the divisor by this term and subtract from the dividend. Repeat until the degree of the new dividend is less than the degree of the divisor.

Start with 1 at the top. Each number is the sum of the two numbers directly above it. The nth row gives the coefficients for (a+b)^(n-1).

Use the formula $(a + b)^n = \sum (n choose k) * a^(n-k) * b^k$, where $(n choose k)$ is the binomial coefficient.

Set the denominator equal to zero and solve for x. Exclude any values that also make the numerator zero (these are holes).

Compare the degrees of the numerator and denominator. If numerator degree < denominator degree, y=0. If numerator degree = denominator degree, y = ratio of leading coefficients. If numerator degree > denominator degree, there is no horizontal asymptote (may have a slant asymptote).

Set x = 0 and evaluate the function.

Set the numerator equal to zero and solve for x. Exclude any values that also make the denominator zero (these are vertical asymptotes or holes).

Factor both the numerator and the denominator. Cancel any common factors.