Notes
Features of polynomials
- A polynomial's degree and leading coefficient set its end behavior. Even degree: both ends go the same way. Odd degree: ends go opposite ways. A positive leading coefficient means the right end rises.
- A degree $n$ polynomial has at most $n$ real zeros and at most $n - 1$ turning points.
- A zero of multiplicity $m$ comes from a factor $(x - r)^m$. Odd multiplicity: the graph crosses. Even multiplicity: it touches and turns.
Solving and factoring
- Factor with a GCF, grouping, sum or difference of cubes, or quadratic form, then use the zero product property.
- Synthetic division by $x - c$ uses $c$. A remainder of 0 means $x - c$ is a factor.
- Remainder Theorem: the remainder of $f(x) \div (x - c)$ is $f(c)$. Factor Theorem: $x - c$ is a factor exactly when $f(c) = 0$.
Graphs and models
- To build a graph: zeros, multiplicities, $y$-intercept $f(0)$, end behavior.
- To write a polynomial from zeros: $f(x) = a(x - r_1)(x - r_2)\cdots$, then use a point to find $a$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Even degree with a negative leading coefficient makes both ends fall.
The zero 1 with multiplicity 2 gives $(x - 1)^2$ and the zero $-3$ gives $(x + 3)$.
The factor $x^2$ has even multiplicity (2), so the graph touches and turns.
$f(3) = 27 - 18 + 4 = 13$.
Use $c = -2$ on $1, 0, -7, -6$: the row is $1, -2, -3$ with remainder 0.
A degree $n$ polynomial has at most $n - 1$ turning points.
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 $f$ by grouping and list its zeros.
Show the answer and scoring guide
$f(x) = x^2(x - 3) - 4(x - 3) = (x - 3)(x^2 - 4) = (x - 3)(x - 2)(x + 2)$. The zeros are $3$, $2$ and $-2$.
Check off the points you earned:
(b) Describe the end behavior and find the $y$-intercept.
Show the answer and scoring guide
The degree is 3 with a positive leading coefficient, so $f$ falls on the left and rises on the right. $f(0) = 12$.
Check off the points you earned:
(c) Find the remainder when $f(x)$ is divided by $x - 1$. Is $x - 1$ a factor?
Show the answer and scoring guide
By the Remainder Theorem, $f(1) = 1 - 3 - 4 + 12 = 6$. The remainder is 6, which is not 0, so $x - 1$ is not a factor.
Check off the points you earned: