Degree and leading coefficient, and how they alone fix a polynomial’s end behavior; zeros, the Factor Theorem, and multiplicity — cross or bounce; long and synthetic division with the Remainder Theorem as the check; the Rational Zeros Theorem; complex numbers and the Fundamental Theorem of Algebra. Then rational functions: domain, vertical and horizontal and slant asymptotes, holes where a factor cancels, and rational equations with their extraneous solutions. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
One rational function, fully factored. The top and the bottom each control something different.
Top gives zeros. Bottom gives asymptotes. A factor that appears on BOTH gives a hole, not an asymptote — so factor everything before you decide which is which.
What people get wrong
⚠️People often think…
In g(x) = 4x − x⁵ + 7 the leading coefficient is 4.
Leading means highest DEGREE, not written first. Put it in standard form: −x⁵ + 4x + 7. The leading coefficient is −1, and that minus decides which way the ends of the graph point.
Standard form first. The highest power leads.
⚠️People often think…
To solve x³ − 4x = 0, divide both sides by x.
That throws away x = 0, which is a real zero. Factor the x out instead: x(x − 2)(x + 2) = 0 gives three zeros — 0, 2 and −2. Dividing by a variable always risks deleting an answer.
Never divide by a variable. Factor it out.
⚠️People often think…
A zero of −5 comes from the factor (x − 5).
The signs flip. The factor (x − 3) has the zero 3, and a zero of −5 comes from (x + 5). Set the factor equal to zero and solve it — do not copy the number across.
Set the factor to zero and solve it.
⚠️People often think…
Every zero of the denominator is a vertical asymptote.
Only the ones that survive canceling. If the same factor sits on top it cancels and leaves a HOLE instead — one missing point, with its height read off the simplified form. Factor, cancel, then decide.
Canceled factor: a hole. Survivor: an asymptote.
Factor first, then read it off
1Watch one
Find all the zeros of x³ − 4x = 0.
Do not divide by x — factor it out.
x(x² − 4) = 0.
The difference of squares splits again: x(x − 2)(x + 2) = 0.
Three zeros: 0, 2 and −2.
2Do one with me
Read f(x) = (x − 2)(x + 3) over (x + 3)(x − 1).
A zero of the function is at x =
A vertical asymptote is at x =
The canceled factor leaves a hole at x =
💬One sentence, then you move on
Why does dividing by x lose one of the zeros?
3Try one
Write the factor that produces a zero of −5.
I want a hint first
You need x plus something to equal zero at −5. Try it both ways and see which one lands there.
💬Last one — then you're done here
What makes a hole different from a vertical asymptote?
Where this goes
Where this lives
Rates that break down at one value: a cost per person when nobody shows up, a speed over a time of zero. The asymptote is where the model stops meaning anything.
What this feeds
Next unit is exponential and logarithmic functions, where growth stops being polynomial.
Name one real quantity that breaks down at a particular value.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🖊️ Sketching a polynomial — the four-step routine
Factor and list the ZEROS with their multiplicities (odd → cross, even → bounce) → read the DEGREE and the sign of the LEADING COEFFICIENT for the end behavior (even: ends match; odd: ends oppose; negative leader flips) → find the y-INTERCEPT f(0) → draw from left to right, turning at most degree − 1 times. Example: f(x) = (x + 1)(x − 2)²: cross at −1, bounce at 2, degree 3 and positive, so down-left and up-right, through (0, 4).
Zeros, ends, intercept, draw.
➗ The whole topic in one idea
Factors are the DNA of a function. A factor (x − c) on top is a zero; a factor on the bottom that survives is a vertical asymptote; a factor that appears on both and cancels is a hole. The leading terms alone decide what happens far away: for a polynomial, end behavior; for a rational function, the horizontal or slant asymptote. Everything else is a plug-in check.
Top gives zeros, bottom gives asymptotes, both give holes.
⚠️ Traps the test loves
Reading the leading coefficient off the first term WRITTEN: g(x) = 4x − x⁵ + 7 leads with −1, not 4. Put it in standard form first.
Dividing by x to solve x³ − 4x = 0 and losing the zero x = 0. Factor x out instead: x(x − 2)(x + 2).
Flipping the sign of a zero: the factor (x − 3) has zero 3, and the zero −5 gives the factor (x + 5).
Calling every zero of the denominator an asymptote. Factor first: a canceled factor is a HOLE, and its height comes from the simplified form.
Using y = 0 as the horizontal asymptote no matter what. Equal degrees → ratio of leading coefficients; bigger degree on top → no horizontal asymptote (maybe a slant one).
Skipping the check after clearing denominators. x = 3 solved the rational equation on paper but made a denominator 0: extraneous, and the equation had no solution.
↗️ End behavior — the four cases
Degree
Leading coefficient
As x → −∞
As x → ∞
Example
even
positive
y → ∞
y → ∞
x², 3x⁴ − x
even
negative
y → −∞
y → −∞
−x² + 4, −2x⁶
odd
positive
y → −∞
y → ∞
x³ − 4x, 5x⁵
odd
negative
y → ∞
y → −∞
−x³ + 2x, 2x − x⁵
🛣️ Asymptotes and holes — the chart
Feature
Where it comes from
Example
Vertical asymptote x = c
a zero of the denominator that does NOT cancel
1/(x − 4): x = 4
Hole at (c, y)
a factor that cancels from top and bottom; y from the simplified form
(x² − 9)/(x − 3): hole at (3, 6)
Horizontal asymptote y = 0
degree on top < degree on bottom
1/(x − 4)
Horizontal asymptote y = a/b
equal degrees; ratio of leading coefficients
(2x² + 1)/(x² − 4): y = 2
Slant asymptote
degree on top is exactly one more; divide and keep the quotient
(x² + 1)/x: y = x
x-intercept
a zero of the numerator (that is not also a zero of the denominator)
(x + 1)/(x − 2): x = −1
🎯 How the test will ask
A polynomial in any order — “degree? leading coefficient? end behavior?” Standard form first, then the four-case chart.
A factored polynomial — “sketch it” or “which graph?” Zeros with cross/bounce, ends, y-intercept.
A graph — “write a possible equation.” Factors from the zeros, multiplicity from the bounces, a from one extra point.
“Divide” or “find the remainder.” Synthetic division for x − c; the Remainder Theorem f(c) as a check.
“Find all zeros” of a cubic with integer coefficients — the p/q list, test until one works, divide, factor the quadratic.
A rational function — “domain, asymptotes, holes, intercepts.” Factor top and bottom before you say anything.
A rational equation — solve, then reject any candidate that zeroes a denominator.
✅ Before the test, can you…
State the degree, leading coefficient and end behavior of any polynomial, even one written out of order?
Find the zeros of x³ − 4x and say which ones cross and which bounce for (x + 1)(x − 2)²?
Divide x³ − 4x + 5 by x − 2 both ways and confirm the remainder 5 with f(2)?
List the rational-zero candidates for 2x³ − 3x² − 3x + 2 and find all three zeros?
Simplify (2 + i)(2 − i) and solve x² + 9 = 0?
Give the vertical asymptote, horizontal asymptote, hole and intercepts of a rational function from its factored form?
Solve a rational equation and explain why a candidate can be extraneous?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: read the end behavior and zeros off a cubic, sort twelve polynomials by what their ends do, name the asymptotes and the hole of a rational function, then put a polynomial long division in order. Every activity checks itself, and hints are free.
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