Revision sheet

Polynomial arithmetic

• Add: combine like terms.
• Multiply: every term by every term; degrees add.

Watch out: $x$-terms add; constants multiply.

End behavior

• Only the leading term (highest power) matters for large $|x|$.
• Even degree: both ends the same way. Odd degree: opposite ways.
• A negative leading coefficient flips the picture.

Watch out: the leading term is not always written first.

Zeros and multiplicity

• $(x - r)$ gives the zero $r$; $(x + q)$ gives the zero $-q$.
• Odd multiplicity: the graph crosses. Even: it touches and turns back.

Watch out: check the sign: $x + 3 = 0$ means $x = -3$.

Polynomial division

$p = d \cdot q + r$; keep 0-coefficients for missing powers.

Watch out: check by multiplying back.

Remainder and factor theorems

• Remainder of $p(x) \div (x - r)$: $p(r)$.
• $x - r$ is a factor $\iff p(r) = 0$.
• For $x + r$, evaluate at $-r$.

Watch out: a factor needs a remainder of $0$.

Polynomial identities

$a^2 - b^2$, $a^3 \pm b^3$, and the geometric-sum identity.

Watch out: the sign pattern in the cube identities.

Asymptotes of rational functions

• Vertical: zeros of the denominator.
• Horizontal: compare degrees. Top smaller: $y = 0$. Equal: ratio of leading coefficients. Top larger: none.

Watch out: use the leading coefficients, not the constants.

Intercepts and holes

1. Cancel common factors: each one leaves a hole.
2. Hole height: the simplified rule at that $x$.
3. $x$-intercepts: remaining zeros of the numerator; $y$-intercept: $f(0)$.

Watch out: zeros of the denominator give asymptotes, not intercepts.

The binomial theorem

• $(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$.
• Coefficients: a row of Pascal's triangle.
• Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$.

Multiplying and dividing rational expressions

Factor, cancel common factors, multiply; divide by multiplying by the reciprocal.

Watch out: cancel factors, never terms.

Adding rational expressions

Common denominator, then add the numerators.

Watch out: never add the denominators.

Rational equations

Clear denominators → solve → check in the original.

Watch out: reject values that make a denominator 0.