Polynomial Functions

A2.B.1End behavior & degree01/04

End behavior and degree.

Step01of 04

Two pieces of a polynomial — its degree (highest power) and its leading coefficient (number on the highest-power term) — fully determine what happens at the far left and far right of its graph.

Even degree, positive leading	→both ends UP
Even degree, negative leading	→both ends DOWN
Odd degree, positive leading	→left DOWN, right UP
Odd degree, negative leading	→left UP, right DOWN

A2.B.1 — End behavior and degree.

Step 01 of 04

Two pieces of a polynomial — its degree (highest power) and its leading coefficient (number on the highest-power term) — fully determine what happens at the far left and far right of its graph.

Even degree, positive leading	→both ends UP
Even degree, negative leading	→both ends DOWN
Odd degree, positive leading	→left DOWN, right UP
Odd degree, negative leading	→left UP, right DOWN

Step 02 of 04

Worked examples. For each, name the end behavior.

$f (x) = 3 x^{4} - 2 x + 1$	→even, positive → both ends up
$g (x) = - x^{3} + 5$	→odd, negative → left up, right down
$h (x) = 2 x^{5}$	→odd, positive → left down, right up
$p (x) = - 4 x^{2} + 7 x$	→even, negative → both ends down

Step 03 of 04

Number of turning points. A polynomial of degree $n$ has AT MOST $n - 1$ turning points (where the graph changes direction).

Degree 2 (parabola)	→at most 1 turning point (the vertex)
Degree 3 (cubic)	→at most 2 turning points
Degree 4 (quartic)	→at most 3 turning points

Could be fewer (e.g., $x^{3}$ has 0 turning points), but never more.

Step 04 of 04

End behavior comes from the leading term alone. All other terms become negligible as $x \to \pm \infty$ . Notation:

f (x) \to + \infty as x \to + \infty

Reads: "as x goes to plus infinity, $f (x)$ also goes to plus infinity."

Key insight

Two pieces of info — degree (even/odd) and sign of leading coefficient — fix the shape at both ends. Even degrees mirror their ends; odd degrees go opposite. Positive leading = right end up; negative = right end down.

A2.B.2 — Zeros, multiplicity, and sketching.

The polynomial

f (x) = (x - 2) (x + 1)^{2} (x - 3)

Step 01 of 04

The zeros (or roots) of $f$ are the x-values where $f (x) = 0$ . From a factored polynomial, read them right off — set each factor to zero.

For our polynomial:

$x - 2 = 0$	→ $x = 2$ (multiplicity 1)
$(x + 1)^{2} = 0$	→ $x = - 1$ (multiplicity 2)
$x - 3 = 0$	→ $x = 3$ (multiplicity 1)

Step 02 of 04

Multiplicity governs graph behavior at the zero.

Odd multiplicity (1, 3, 5, ...)	→the graph CROSSES the x-axis at that zero
Even multiplicity (2, 4, 6, ...)	→the graph TOUCHES the x-axis and bounces back
Higher odd (3, 5)	→crosses, but flattens through
Higher even (4, 6)	→touches, but flattens

Our polynomial: crosses at $x = 2$ and $x = 3$ (multiplicity 1), bounces at $x = - 1$ (multiplicity 2).

Step 03 of 04

Sketch the polynomial. Combine zeros, multiplicity, and end behavior. Degree is $1 + 2 + 1 = 4$ (even) and leading coefficient is positive — both ends UP.

Crosses at x = 2, 3. Bounces at x = -1 (multiplicity 2).

Step 04 of 04

Total roots, counting multiplicity = degree. A polynomial of degree $n$ has exactly $n$ zeros when you count each by its multiplicity (and allow complex zeros). This is the Fundamental Theorem of Algebra.

Our degree-4 polynomial has zeros 2, -1, -1, 3 — counting the double root twice gives 4 zeros total.

Key insight

Each factor in a polynomial gives a zero. The exponent on the factor = multiplicity. Odd multiplicity crosses the axis; even multiplicity touches and bounces. Total multiplicities equal the degree.

A2.B.3 — Remainder and factor theorems.

Step 01 of 04

Remainder theorem. When you divide $f (x)$ by $x - c$ , the remainder is $f (c)$ . No long division needed — just plug in.

Remainder of f (x) \div (x - c) = f (c)

Example: $f (x) = x^{3} - 4 x + 1$ divided by $x - 2$ :

f (2) = 8 - 8 + 1 = 1 (the remainder)

Step 02 of 04

Factor theorem. The special case where the remainder is zero:

(x - c) is a factor of f ⟺ f (c) = 0

In words: $c$ is a zero of $f$ if and only if $x - c$ is a factor of $f$ . Two ways of saying the same thing.

Step 03 of 04

Use the factor theorem to test possible factors. For $f (x) = x^{3} - 7 x + 6$ , is $x - 1$ a factor?

f (1) = 1 - 7 + 6 = 0 ✓

Yes. So $x - 1$ is a factor, and we can divide it out:

f (x) = (x - 1) (x^{2} + x - 6) = (x - 1) (x + 3) (x - 2)

Zeros: $1, - 3, 2$ .

Step 04 of 04

Why this matters. Pairing the factor theorem with synthetic division gives a workflow for fully factoring higher-degree polynomials:

1.	→guess a rational zero (next skill helps narrow the list)
2.	→verify with f(c) = 0
3.	→synthetic-divide to get a smaller polynomial
4.	→repeat until the quotient is quadratic, then factor or use the quadratic formula

Key insight

Remainder theorem: dividing by

x - c

gives remainder

f (c)

. Factor theorem: zero remainder ⟺

x - c

is a factor ⟺

c

is a zero. Three ways of saying one fact.

A2.B.4 — Rational root theorem.

Step 01 of 04

Rational root theorem. Suppose $f (x) = a_{n} x^{n} + \dots + a_{0}$ has integer coefficients. ANY rational zero is of the form $p / q$ where:

$p$	→a factor of the constant term $a_{0}$
$q$	→a factor of the leading coefficient $a_{n}$

The theorem doesn't say a rational zero exists — only that IF one exists, it's on this finite list.

Step 02 of 04

Worked example. List candidates for $f (x) = 2 x^{3} + 3 x^{2} - 8 x + 3$ .

Constant term $a_{0} = 3$ , factors $p$ : $\pm 1, \pm 3$ . Leading coefficient $a_{n} = 2$ , factors $q$ : $\pm 1, \pm 2$ .

Candidates $p / q$ : $\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}$ . Eight candidates total.

Step 03 of 04

Test each candidate by plugging in (factor theorem). Start with small / clean ones.

f (1) = 2 + 3 - 8 + 3 = 0 ✓

Found one. Synthetic-divide by $x - 1$ :

f (x) = (x - 1) (2 x^{2} + 5 x - 3)

Factor the quadratic: $2 x^{2} + 5 x - 3 = (2 x - 1) (x + 3)$ . Final factorization:

f (x) = (x - 1) (2 x - 1) (x + 3)

zeros: x = 1, \frac{1}{2}, - 3

Step 04 of 04

What if no rational zero works? The polynomial may have only irrational or complex zeros. The rational root theorem then comes back empty — that's not a failure, it's a valid result. Use other techniques (numerical methods, complex factoring) to find the remaining zeros.

Key insight

Rational root theorem narrows an infinite candidate list to a finite one:

p / q

with

p

dividing the constant and

q

dividing the leading coefficient. Test each, keep the ones that give zero, divide them out, repeat on the smaller polynomial.

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A2.B.1 – A2.B.4 · CCSS F.IF.C.7c / A.APR.B.3 / A.APR.B.2·Full curriculum →