Notes
Degree and end behavior
The leading term decides the ends of the graph.
| Degree | Leading coefficient $> 0$ | Leading coefficient $< 0$ |
|---|---|---|
| Even | up on both ends | down on both ends |
| Odd | down left, up right | up left, down right |
Operations and division
- Subtracting: distribute the minus to every term. Multiplying: every term times every term.
- Synthetic division by $x - c$ uses $c$; the last number is the remainder. Use long division for other divisors.
- Remainder Theorem: dividing $p(x)$ by $x - c$ leaves $p(c)$. Factor Theorem: $x - c$ is a factor exactly when $p(c) = 0$.
Zeros
- Fundamental Theorem of Algebra: degree $n$ means $n$ complex zeros, counting multiplicity.
- Nonreal zeros of real polynomials come in conjugate pairs $a \pm bi$.
- Rational Root Theorem: possible rational zeros are $\displaystyle \pm\frac{p}{q}$, with $p$ a factor of the constant and $q$ a factor of the leading coefficient.
Graphing
- Odd multiplicity: crosses the axis. Even multiplicity: touches and turns.
- At most $n - 1$ turning points; $y$-intercept is $f(0)$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The leading term $x^4$ has even degree and positive coefficient.
Synthetic with $-2$: $1$; $2 - 2 = 0$; $-1 + 0 = -1$; $-2 + 2 = 0$.
$p(-1) = -2 + 1 + 5 = 4$.
$2i$ must also be a zero, giving at least three zeros: $4$, $-2i$, $2i$.
Factors of 3 over factors of 2.
$(x^2 - 4)(x^2 + 4) = 0$, so $x^2 = 4$ or $x^2 = -4$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A box with no lid is made from a 16-inch by 10-inch sheet of cardboard by cutting a square with side $x$ inches from each corner and folding up the sides. Its volume is $V(x) = x(16 - 2x)(10 - 2x)$ cubic inches.
(a) Write $V(x)$ in standard form and state the values of $x$ that make sense for the box.
Show the answer and scoring guide
$(16 - 2x)(10 - 2x) = 4x^2 - 52x + 160$, so $V(x) = 4x^3 - 52x^2 + 160x$. The width $10 - 2x$ must be positive, so $0 < x < 5$.
Check off the points you earned:
(b) Find $V(2)$ and describe the box.
Show the answer and scoring guide
$V(2) = 2(12)(6) = 144$. The box is 12 in by 6 in by 2 in, with volume 144 cubic inches.
Check off the points you earned:
(c) Use synthetic division to find every $x$ with $V(x) = 144$. Which answers make sense?
Show the answer and scoring guide
$4x^3 - 52x^2 + 160x - 144 = 0 \Rightarrow x^3 - 13x^2 + 40x - 36 = 0$. Dividing by $x - 2$ gives $x^2 - 11x + 18 = (x - 2)(x - 9)$. So $x = 2$ (a double root) or $x = 9$. Only $x = 2$ is in the domain.
Check off the points you earned: