Polynomial Facts

Polynomial Facts

Facts about polynomials of the form p(x) = a_nxⁿ + a_n_–1xⁿ^–1 + ··· + a₂x² + a₁x + a₀ are listed below.

Polynomial End Behavior:
1. If the degree n of a polynomial is even, then the arms of the graph are either both up or both down.
2. If the degree n is odd, then one arm of the graph is up and one is down.
3. If the leading coefficient a_n is positive, the right arm of the graph is up.
4. If the leading coefficient a_n is negative, the right arm of the graph is down.

Extreme Values:
The graph of a polynomial of degree n has at most n – 1 extreme values.

Inflection Points:
The graph of a polynomial of degree n has at most n – 2 inflection points.

Remainder Theorem:
p(c) is the remainder when polynomial p(x) is divided by x – c.

Factor Theorem:
x – c is a factor of polynomial p(x) if and only if c is a zero of p(x).

Rational Root Theorem:
If a polynomial equation a_nxⁿ + a_n_–1xⁿ^–1 + ··· + a₂x² + a₁x + a₀ = 0 has integer coefficients then it is possible to make a complete list of all possible rational roots. This list consists of all possible numbers of the form c/d, where c is any integer that divides evenly into the constant term a₀ and d is any integer that divides evenly into the leading term a_n.

Conjugate Pair Theorem:
If a polynomial has real coefficients then any complex zeros occur in complex conjugate pairs. That is, if a + bi is a zero then so is a – bi, where a and b are real numbers.

Fundamental Theorem of Algebra:
A polynomial p(x) = a_nxⁿ + a_n_–1xⁿ^–1 + ··· + a₂x² + a₁x + a₀ of degree at least 1 and with coefficients that may be real or complex must have a factor of the form x – r, where r may be real or complex.

Corollary of the Fundamental Theorem of Algebra:
A polynomial of degree n must have exactly n zeros, counting mulitplicity.

Key Formula

p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0

Where:

$p(x)$ = The polynomial function
$n$ = The degree of the polynomial (highest power of x)
$a_n$ = The leading coefficient (coefficient of the highest-degree term, aₙ ≠ 0)
$a_0$ = The constant term
$x$ = The variable

Worked Example

Problem: Use the Rational Root Theorem and Factor Theorem to find all zeros of p(x) = 2x³ + 3x² − 8x + 3.

Step 1: Identify the constant term a₀ = 3 and the leading coefficient aₙ = 2. List divisors of each: divisors of 3 are ±1, ±3; divisors of 2 are ±1, ±2.

\text{Possible rational roots} = \pm\frac{\{1, 3\}}{\{1, 2\}} = \pm 1,\; \pm 3,\; \pm\tfrac{1}{2},\; \pm\tfrac{3}{2}

Step 2: Test x = 1 using the Remainder Theorem: evaluate p(1).

p(1) = 2(1)^3 + 3(1)^2 - 8(1) + 3 = 2 + 3 - 8 + 3 = 0

Step 3: Since p(1) = 0, the Factor Theorem tells us (x − 1) is a factor. Divide p(x) by (x − 1) using synthetic or long division.

2x^3 + 3x^2 - 8x + 3 = (x - 1)(2x^2 + 5x - 3)

Step 4: Factor the quadratic 2x² + 5x − 3. Find two numbers that multiply to (2)(−3) = −6 and add to 5: those are 6 and −1.

2x^2 + 5x - 3 = (2x - 1)(x + 3)

Step 5: Set each factor equal to zero to find all zeros.

x - 1 = 0 \Rightarrow x = 1, \quad 2x - 1 = 0 \Rightarrow x = \tfrac{1}{2}, \quad x + 3 = 0 \Rightarrow x = -3

Answer: The three zeros of p(x) are x = 1, x = 1/2, and x = −3. This confirms the Fundamental Theorem corollary: a degree-3 polynomial has exactly 3 zeros.

Another Example

This example focuses on end behavior, extreme value/inflection point bounds, and the Conjugate Pair Theorem — contrasting with the first example, which focused on the Rational Root Theorem, Remainder Theorem, and Factor Theorem.

Problem: Determine the end behavior, maximum number of extreme values, and maximum number of inflection points for f(x) = −3x⁴ + 2x³ − x + 7. Also, if 2 + i is a zero, identify another zero.

Step 1: The degree is n = 4 (even) and the leading coefficient is aₙ = −3 (negative). Since the degree is even, both arms point the same direction. Since the leading coefficient is negative, the right arm points down. Therefore both arms point down.

\text{As } x \to \pm\infty,\; f(x) \to -\infty

Step 2: Maximum number of extreme values: n − 1 = 4 − 1 = 3. Maximum number of inflection points: n − 2 = 4 − 2 = 2.

\text{At most } 3 \text{ extreme values and } 2 \text{ inflection points}

Step 3: Apply the Conjugate Pair Theorem. The polynomial has all real coefficients, so complex zeros come in conjugate pairs. If 2 + i is a zero, then 2 − i must also be a zero.

\text{If } 2 + i \text{ is a zero, then } 2 - i \text{ is also a zero.}

Step 4: By the Fundamental Theorem corollary, f(x) has exactly 4 zeros (counting multiplicity). Two are 2 + i and 2 − i, so the remaining two zeros must be real.

\text{4 total zeros} - 2 \text{ complex zeros} = 2 \text{ real zeros remaining}

Answer: f(x) has both arms pointing down, at most 3 extreme values, at most 2 inflection points, and 2 − i is guaranteed to be a zero alongside 2 + i.

Frequently Asked Questions

How do you determine the end behavior of a polynomial?

Look at two things: the degree and the leading coefficient. If the degree is even, both ends of the graph go the same direction; if odd, they go opposite directions. The sign of the leading coefficient tells you the direction of the right arm — positive means up, negative means down.

What is the difference between the Remainder Theorem and the Factor Theorem?

The Remainder Theorem says that when you divide p(x) by (x − c), the remainder equals p(c). The Factor Theorem is a special case: if that remainder is zero — meaning p(c) = 0 — then (x − c) is a factor of p(x). So the Factor Theorem identifies factors, while the Remainder Theorem gives you any remainder.

How many zeros does a polynomial have?

By the Fundamental Theorem of Algebra and its corollary, a polynomial of degree n has exactly n zeros when you count multiplicity and include complex zeros. For example, a degree-5 polynomial always has exactly 5 zeros total, though some may be repeated or complex.

Remainder Theorem vs. Factor Theorem

	Remainder Theorem	Factor Theorem
What it states	The remainder when p(x) is divided by (x − c) equals p(c)	(x − c) is a factor of p(x) if and only if p(c) = 0
What it finds	The numerical remainder of polynomial division	Whether a given value is a zero/root of the polynomial
Relationship	General statement about any division by (x − c)	Special case of the Remainder Theorem when the remainder is 0
Typical use	Evaluating p(c) quickly without full substitution	Confirming a root and then factoring the polynomial

Why It Matters

These polynomial facts appear throughout Algebra 2, Precalculus, and AP Calculus courses. You need them to factor higher-degree polynomials, sketch their graphs accurately, and solve polynomial equations. On standardized tests like the SAT and ACT, questions about end behavior, finding zeros, and counting roots rely directly on these theorems.

Common Mistakes

Mistake: Confusing the maximum number of extreme values with the degree itself.

Correction: A degree-n polynomial has at most n − 1 extreme values (local maxima and minima), not n. For example, a cubic (degree 3) has at most 2 extreme values.

Mistake: Forgetting to include both conjugates when a complex zero is found.

Correction: The Conjugate Pair Theorem applies whenever the polynomial has real coefficients. If 3 − 2i is a zero, you must include 3 + 2i as well. Omitting it leads to incorrect factorizations and a wrong count of zeros.

Related Terms

Polynomial — The type of function these facts describe
Remainder Theorem — Gives the remainder when dividing by (x − c)
Factor Theorem — Identifies factors from zeros of the polynomial
Rational Root Theorem — Lists all possible rational roots to test
Fundamental Theorem of Algebra — Guarantees every polynomial has at least one root
Conjugate Pair Theorem — Complex zeros of real polynomials come in pairs
End Behavior — Describes how the graph behaves as x → ±∞
Extreme Values of a Polynomial — Local maxima and minima bounded by degree − 1