Notes
Polynomials
- The leading term sets end behavior. Even degree: both ends match; odd degree: ends go opposite ways. The sign of the leading coefficient picks up or down.
- A factor $(x - r)^m$ gives a zero of multiplicity $m$. Odd $m$ crosses the axis, even $m$ touches and turns.
- Remainder Theorem: dividing by $x - c$ leaves remainder $f(c)$. If $f(c) = 0$, then $x - c$ is a factor.
- Rational zeros have the form $\displaystyle \frac{p}{q}$ ($p$ divides the constant, $q$ divides the leading coefficient). Complex zeros of real polynomials come in conjugate pairs.
Rational functions
- Vertical asymptotes come from denominator zeros that do not cancel; canceled factors make holes.
- Horizontal asymptote: compare degrees. Smaller top gives $y = 0$; equal degrees give the ratio of leading coefficients; a top that is one degree larger gives a slant asymptote.
Inequalities
- Find zeros and undefined points, then test each interval with a sign chart. Never include values that make a denominator zero.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Even degree with a negative leading coefficient.
The factor $x^2$ has even multiplicity.
$f(-1) = -2 - 1 + 4 = 1$.
It factors as $\displaystyle \frac{(x - 1)(x + 1)}{(x - 2)(x + 1)}$, so $x + 1$ cancels (hole) and $x - 2$ stays (asymptote).
$(x - 3)(x + 2) < 0$ between the zeros, with endpoints excluded.
Conjugates come in pairs, so $1 - i$ must be a zero.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $f(x) = x^3 - 3x^2 - 4x + 12$.
(a) Factor $f$ completely and list its zeros.
Show the answer and scoring guide
Group: $x^2(x - 3) - 4(x - 3) = (x - 3)(x^2 - 4) = (x - 3)(x - 2)(x + 2)$. The zeros are $3$, $2$ and $-2$.
Check off the points you earned:
(b) Describe the end behavior and give the y-intercept.
Show the answer and scoring guide
Degree 3 with a positive leading coefficient: down on the left, up on the right. The y-intercept is $f(0) = 12$.
Check off the points you earned:
(c) Solve $f(x) > 0$.
Show the answer and scoring guide
The sign changes at $-2$, $2$ and $3$. For $x > 3$ it is positive, between 2 and 3 negative, between $-2$ and 2 positive, below $-2$ negative. Solution: $(-2, 2) \cup (3, \infty)$.
Check off the points you earned: