Revision sheet

Polynomial arithmetic

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

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

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

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$.

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 $0$.

All zeros from one zero

1. Check $p(r) = 0$.
2. Divide by $x - r$: $p(x) = (x - r)\,q(x)$.
3. Solve $q(x) = 0$; list $r$ with those solutions.

Watch out: $x^2 = 9$ gives $3$ and $-3$.

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.

Complex zeros

• $i^2 = -1$, so $\sqrt{-k^2} = ki$.
• $b^2 - 4ac < 0$: no real zeros, two complex ones.
• Real coefficients: $p + qi$ and $p - qi$ come together.

Watch out: the conjugate changes only the sign of the $i$ part.