Unit 6 is about polynomial functions: how to read a graph from an equation, how to divide polynomials, and how many roots a polynomial has. By the end, you should be able to sketch a polynomial from its factored form, use synthetic division and the Remainder Theorem to find factors, and list every zero of a polynomial, including complex ones.
Read the overview first, then work through each topic's notes and checkpoint question before checking the common mistakes at the bottom.
Polynomials are the first big family of functions in Algebra II beyond lines and parabolas. This unit moves from graphs (6.1) to division tools (6.2) to the theory of how many roots exist and where they come from (6.3).
Honors Algebra II Unit 6 covers polynomial functions and the theory behind their roots. You learn how the degree and leading coefficient set end behavior, how multiplicity decides whether a graph crosses or touches the x-axis, and how to divide polynomials with long division and synthetic division. The unit then connects division to the Remainder Theorem and Factor Theorem, uses the Rational Root Theorem to find candidate zeros, and finishes with the Fundamental Theorem of Algebra, conjugate pairs and Vieta's formulas.
For a polynomial of positive degree, even degree means both ends point the same way, and odd degree means they point opposite ways. A positive leading coefficient makes the right end rise. A degree- polynomial with has at most turning points. A nonzero constant polynomial stays at the same value and has no turning points.
The Remainder Theorem says the remainder from dividing by equals . If that remainder is 0, then is a factor, and the quotient is a smaller polynomial you can keep factoring.
Some of those zeros may be real, some may repeat, and some may be non-real complex numbers. If the coefficients are real, non-real zeros show up in conjugate pairs like and .
If , then is a factor of , and if is real, the graph hits the x-axis at . Graphing, dividing and root-finding are all ways of moving between these three views. Once you see that, the theorems in this unit read like shortcuts instead of separate facts to memorize.
A polynomial function is a sum of terms like a x^n where every exponent is a whole number.
Polynomial long division works like number long division.
The Fundamental Theorem of Algebra says every non-constant polynomial has at least one complex root.
A polynomial function is a sum of terms like where every exponent is a whole number. So is a polynomial, but or terms are not allowed.
Two numbers control the big picture. For a polynomial of positive degree, the degree (highest exponent) tells you whether the ends match, and the leading coefficient tells you which way the right end goes. A nonzero constant polynomial stays at the same value. Zeros give the x-intercepts, and each zero's multiplicity tells you what happens there. Odd multiplicity means the graph crosses the axis. Even multiplicity means it touches and turns back.
Don't assume symmetry from the degree. is odd-degree but has no symmetry, because it mixes odd and even powers.
Worked example. Sketch . Adding exponents gives degree , and the leading coefficient is . Odd degree with a negative leading coefficient means the graph rises on the left and falls on the right. The zero has multiplicity 2, so the graph touches at . The zero has multiplicity 1, so it crosses at . The y-intercept is . With 3 as the degree, there are at most 2 turning points, which matches a curve that comes down from the upper left, crosses at , dips to a low point, rises to touch at , and then falls.
Check yourself: What is the end behavior of , and how many turning points can it have at most? (Answer: Even degree with a negative leading coefficient means both ends fall, and it can have at most 3 turning points.)
| Degree | Leading coefficient | Left end (x to $-\infty$) | Right end (x to $+\infty$) | Example |
|---|---|---|---|---|
| Even | Positive | Rises | Rises | $x^2$, $x^4$ |
| Even | Negative | Falls | Falls | $-x^2$, $-x^4$ |
| Odd | Positive | Falls | Rises | $x^3$ |
| Odd | Negative | Rises | Falls | $-x^3$ |
Polynomial long division works like number long division. You divide leading terms, multiply, subtract, and bring down the next term until the remainder has a lower degree than the divisor. The result always fits .
When the divisor has the form , synthetic division does the same job with just the coefficients. Write a 0 for any missing power, or the columns line up wrong.
The Remainder Theorem says dividing by leaves remainder . The Factor Theorem follows from it, so is a factor exactly when . Watch the sign. To test , plug in .
Worked example. Factor , given that is a zero. Use synthetic division with and coefficients . Bring down the 1. Then , and . Next , and . Last , and . The remainder is 0, so is a factor and the quotient is . That makes , with zeros , and . You can confirm with the Remainder Theorem, since .
Check yourself: What is the remainder when is divided by ? (Answer: , so the remainder is 2.)
| Method | When you can use it | What you get |
|---|---|---|
| Long division | Any polynomial divisor, such as $x^2 + 1$ | Quotient and remainder |
| Synthetic division | Only linear divisors of the form $(x - c)$ | Quotient coefficients and remainder |
| Remainder Theorem | Only linear divisors of the form $(x - c)$ | The remainder only, found by computing $P(c)$ |
The Fundamental Theorem of Algebra says every non-constant polynomial has at least one complex root. Applying that repeatedly gives the version you'll use most often. A degree- polynomial has exactly complex roots when you count multiplicity. The theorem says the roots exist, but it doesn't tell you how to find them.
For a polynomial with integer coefficients, you can use the Rational Root Theorem to look for rational roots. Any rational root in lowest terms has dividing the constant term and dividing the leading coefficient. If the constant term is 0, factor out first. Otherwise, list and test the candidates, then divide out any roots you find. If you reach a quadratic, use the quadratic formula. If no rational candidate works, you need another method, such as numerical approximation.
If the coefficients are real, non-real roots come in conjugate pairs. That is why a real cubic always has at least one real root. Vieta's formulas link roots to coefficients. For , the sum of the roots is and the product is .
Worked example. Find all zeros of . The candidates are . Testing gives . Synthetic division by gives the quotient . Setting gives . The zeros are , and . That is three zeros for degree 3, with the non-real zeros forming a conjugate pair.
Check yourself: A degree-4 polynomial with real coefficients has roots , and . What is the fourth root? (Answer: , because non-real roots of real-coefficient polynomials come in conjugate pairs.)
| Term | Definition |
|---|---|
| Polynomial function | A function made of terms $a x^n$ with whole-number exponents, combined by addition and subtraction. |
| End behavior | What $f(x)$ does as $x$ goes to $+\infty$ and $-\infty$, set by the degree and leading coefficient. |
| Zero (root) | A value $c$ where $f(c) = 0$, which appears as an x-intercept when $c$ is real. |
| Multiplicity | How many times a factor $(x - c)$ appears in the fully factored polynomial. |
| Synthetic division | A coefficient-only shortcut for dividing a polynomial by $(x - c)$. |
| Remainder Theorem | The remainder from dividing $P(x)$ by $(x - c)$ equals $P(c)$. |
| Factor Theorem | $(x - c)$ is a factor of $P(x)$ exactly when $P(c) = 0$. |
| Rational Root Theorem | For a polynomial with integer coefficients and a nonzero constant term, a rule that lists possible rational roots as factors of the constant term over factors of the leading coefficient. If the constant term is 0, factor out $x$ first. |
| Fundamental Theorem of Algebra | A degree-$n$ polynomial has exactly $n$ complex roots, counting multiplicity. |
| Complex conjugate | The partner $a - bi$ of a complex number $a + bi$, which is also a root whenever a real-coefficient polynomial has $a + bi$ as a root. |
The factor means , so the matching zero is , not . Before using synthetic division or the Remainder Theorem, rewrite the factor as and use that .
For , there's no term, so the coefficient row should be . Leaving out the 0 misaligns the coefficients, so the quotient or remainder can come out wrong.
The theorem only lists possible rational roots, and many of them won't work. Each candidate needs to be tested by substitution or synthetic division, and some polynomials have no rational roots at all.
A degree- polynomial has complex roots, but some may be non-real or repeated, so it can have fewer x-intercepts. For example, has none. For , the maximum number of turning points is , but has zero of them.