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What Is a Polynomial?

A polynomial is an algebraic expression made from constants, variables, and addition, subtraction, or multiplication. The variable’s exponents must be nonnegative whole numbers.

For example,

\[ 4x^3-2x+9 \]

is a polynomial. Its terms are $4x^3$, $-2x$, and $9$. The numbers $4$ and $-2$ are coefficients, while $9$ is a constant.

Expressions such as $x^{-1}+2$, $\sqrt{x}+1$, and $\frac{3}{x}$ are not polynomials because they use negative, fractional, or variable exponents.

A polynomial expression does not contain an equals sign. A polynomial equation does, such as

\[ x^2-7x+12 = 0. \]

Types of Polynomials

You can classify a polynomial by its degree, which is the highest exponent after simplifying.

A constant has degree $0$, such as $8$.
A linear polynomial has degree $1$, such as $3x-2$.
A quadratic has degree $2$, such as $x^2+5x+6$.
A cubic polynomial has degree $3$.
A quartic polynomial has degree $4$.
A quintic has degree $5$.
Any polynomial of degree $6$ or above is a higher-degree polynomial.

You can also classify it by its number of terms:

A monomial has one term.
A binomial has two terms.
A trinomial has three terms.
A polynomial with four or more terms is usually just called a polynomial.

These are different descriptions. For example,

\[ 7x^5-3x^2+4x-11 \]

has four terms, but its degree is $5$. Missing powers do not change this. A polynomial does not need every exponent between its highest and lowest powers.

How to Simplify Polynomials

To simplify polynomials, combine like terms. Like terms have the same variable part and the same exponent

\[ 3x^2+5x-2x^2+7 = x^2+5x+7 \]

The $3x^2$ and $-2x^2$ combine because both are $x^2$-terms. The $5x$ and $7$ cannot combine with them.

Writing terms from greatest exponent to least exponent gives standard form. Simplifying is not factoring: simplification combines terms, while factoring rewrites an expression as a product.

How to Evaluate a Polynomial

Briefly explain substituting a value for x and calculating $P(x)$.

Example:

\[ P(x) = 2x^3-3x+5 \]

To find (P(2)):

\[ P(2) = 2(2)^3-3(2)+5 \]

\[ =16-6+5=15 \]

Evaluating a polynomial means finding its value for a specific input. This is also useful when checking possible roots with the Factor Theorem or graphing a polynomial function.

How to Add and Subtract Polynomials

Adding Polynomials

When you add polynomials, remove the parentheses and combine matching terms.

\[ (4x^3-3x^2+7)+(2x^3+5x^2-4) \]

\[ =6x^3+2x^2+3 \]

Only terms with identical variable parts combine. This is the main idea behind an adding and subtracting polynomials calculator.

Subtracting Polynomials

For subtraction, distribute the negative sign through every term in the second polynomial before combining.

\[ (5x^2-2x+6)-(3x^2+4x-1) \]

\[ =5x^2-2x+6-3x^2-4x+1 \]

\[ =2x^2-6x+7 \]

A common subtracting-polynomials mistake is changing only the first sign inside the second parentheses.

How to Multiply Polynomials

Multiplication uses the distributive property: every term in one factor must multiply every term in the other.

\[ (2x-3)(x^2+4x-5) \]

\[ =2x(x^2+4x-5)-3(x^2+4x-5) \]

\[ =2x^3+8x^2-10x-3x^2-12x+15 \]

\[ =2x^3+5x^2-22x+15 \]

FOIL is a convenient shortcut when multiplying two binomials. It is not the general method for larger polynomial expressions. To multiply polynomials correctly, make sure no required pair of terms is skipped.

How to Divide Polynomials

If every term shares a monomial factor, divide each term separately:

\[ \frac{12x^3-6x^2+18x}{6x}=2x^2-x+3. \]

For division by another polynomial, use polynomial long division. For example,

\[ \frac{x^3+2x^2-5x-6}{x+2}=x^2-5+\frac{4}{x+2}. \]

The quotient is $x^2-5$, and the remainder is $4$. You can check this by multiplying:

\[ (x+2)(x^2-5)+4=x^3+2x^2-5x-6. \]

Polynomial long division works when the divisor is any nonzero polynomial, while synthetic division is a shortcut for linear divisors of the form $x−a$. For divisors with degree 2 or higher, use polynomial long division or another appropriate division method.

How to Factor Polynomials

Factoring rewrites a polynomial as a product of simpler factors. Start by checking for a greatest common factor.

Useful patterns include:

Difference of squares: $\,a^2-b^2=(a-b)(a+b)$
Perfect-square trinomials, such as $\,x^2+6x+9=(x+3)^2$
Trinomials, such as $\,x^2-7x+12=(x-3)(x-4)$
Grouping, which can reveal a shared binomial factor

Not every polynomial factors over the integers. Some factor only using irrational or complex numbers, and some have no convenient factored form in the number system you are using. For more focused practice with these patterns, see the Factoring Calculator.

How to Solve Polynomial Equations

Solving means finding variable values that make an equation true. First, make sure you have an equation, not only an expression.

For example,

\[ x^2-7x+12=0 \]

can be factored:

\[ (x-3)(x-4)=0. \]

By the zero-product property,

\[ x=3 \quad \text{or} \quad x=4. \]

Factoring is useful, but it is not universal. A quadratic may need the quadratic formula. A cubic or higher-degree equation may use possible rational roots, synthetic division, polynomial long division, or a numerical approximation when no simple exact factorization exists. For quadratic-specific help, use the Quadratic Equation Calculator.

Polynomial Roots and Zeros

Roots are values that solve a polynomial equation. Zeros are inputs that make a polynomial function equal zero. In practice, they refer to the same values when you set the function equal to zero.

For

\[ f(x)=x^2-5x+6, \]

set the function equal to zero:

\[ x^2-5x+6=0. \]

Then,

\[ (x-2)(x-3)=0, \]

so the zeros, or roots, are $2$ and $3$.

Common Types of Roots include real roots, complex roots, and repeated roots. A repeated root has multiplicity greater than $1$; for example, $x=2$ has multiplicity $2$ in $(x-2)^2$.

The multiplicity of a root also affects the graph: roots with odd multiplicity generally cross the x-axis, while roots with even multiplicity generally touch the x-axis and turn around.

The Fundamental Theorem of Algebra says a degree-$n$ polynomial has $n$ complex roots when multiplicity is counted. Those roots are not necessarily distinct, rational, or real.

Degree, Terms, and Polynomial Form

Consider again:

\[ 7x^5-3x^2+4x-11. \]

Its degree is $5$, because $5$ is the largest exponent. Its leading coefficient is $7$, and its constant term is $-11$.

To find the number of terms in this polynomial, separate it at addition and subtraction signs:

\[ 7x^5,\quad -3x^2,\quad 4x,\quad -11. \]

It has four terms. Degree measures the highest power; term count measures the number of separate pieces.

Standard form arranges terms in descending powers. Expanded form shows multiplication carried out, while factored form displays factors. For instance,

\[ x^2-7x+12 \]

is expanded, while

\[ (x-3)(x-4) \]

is factored.

Polynomial Functions and Graphs

A polynomial function assigns one output to each input. For example,

\[ f(x)=2x^3-5x+1 \]

is a polynomial function because each chosen $x$-value produces one value of $f(x)$.

Its graph can help you see the real zeros, which occur where the graph intersects the x-axis. The degree and leading coefficient also influence general end behavior. A cubic with a positive leading coefficient, for instance, tends downward on the far left and upward on the far right.

A polynomial function calculator is especially useful when you want both algebraic work and a graph of a suitable entered function.

Worked Example

Solve:

\[ x^3-2x^2-5x+6=0. \]

\[ P(1)=1^3-2(1)^2-5(1)+6=0. \]
Since $P(1)=0$, the Factor Theorem says $x-1$ is a factor.
\[ x^3-2x^2-5x+6=(x-1)(x^2-x-6). \]
Synthetic division by $x-1$ leaves the quadratic $x^2-x-6$.
\[ x^2-x-6=(x-3)(x+2). \]
The quadratic factors because $-3\cdot2=-6$ and $-3+2=-1$.
\[ (x-1)(x-3)(x+2)=0. \]
A product is zero when at least one factor is zero.
\[ x=1,\quad x=3,\quad x=-2. \]
These are the three real roots of this cubic polynomial.
\[ P(3)=27-18-15+6=0. \]
Substitution verifies that $3$ is a solution; the other two can be checked the same way.

You could also use polynomial long division after finding a root, but synthetic division is shorter here because the divisor is linear.

Common Mistakes

Combining unlike terms, such as treating $3x^2+4x$ as $7x^3$
Forgetting to distribute a negative through every term during subtraction
Multiplying only some terms instead of every required pair
Using FOIL for products that are not two binomials
Losing signs while subtracting during polynomial long division
Using synthetic division when the divisor is not in the form $x-a$
Confusing the degree of a polynomial with its number of terms
Assuming every polynomial factors over the integers
Trying to solve an expression before setting it equal to zero
Stopping factoring too early: Some polynomial expressions can be factored further after the first factoring step.

Questions

Polynomial Questions Students Ask

What is a polynomial?

A polynomial is an expression made from constants and variables with nonnegative whole-number exponents. For example, $3x^2-4x+7$ is a polynomial, but $x^{-1}+2$ is not.

How do you add polynomials?

Remove parentheses and combine only like terms. For example, \[ (2x^2+3x)+(5x^2-x)=7x^2+2x. \]

How do you subtract polynomials?

Distribute the subtraction sign across every term in the second polynomial, then combine like terms. For example, \[ (4x^2+x)-(x^2-3x)=3x^2+4x. \]

How do you multiply polynomials?

Use the distributive property so every term in one polynomial multiplies every term in the other. Then combine like terms in the expanded result.

How do you divide polynomials?

Divide by a monomial term-by-term when possible. Otherwise, use polynomial long division, or synthetic division when the divisor has the form $x-a$.

How do you factor a polynomial?

First check for a greatest common factor. Then look for patterns such as a difference of squares, a trinomial, grouping, or a perfect-square trinomial.

How do you solve a polynomial equation?

Set the equation equal to zero and choose a suitable method. Factoring, the quadratic formula, synthetic division, and numerical methods can all be appropriate depending on the polynomial.

How do you find the degree of a polynomial?

Simplify first, then identify the greatest exponent of the variable. In $6x^4-x^2+9$, the degree is $4$.

What are the roots of a polynomial?

Roots are values of the variable that make a polynomial equation equal zero. They may be real or complex, and a root can repeat.

Can every polynomial be factored?

No. A polynomial may not factor over the integers or rationals. It may factor over the real or complex numbers instead, or require numerical approximation for its roots.

How do you find the number of terms in a polynomial?

To find the number of terms, first simplify the polynomial by combining all like terms. Then, count each distinct monomial block separated by addition or subtraction operators.

What is the difference between polynomial roots and zeros?

Zeros refer specifically to inputs of a function $f(x)$ where $f(x) = 0$. Roots refer to the solution values of an equation $P(x) = 0$. Mathematically, the zeros of $f(x)$ are identical to the roots of $f(x) = 0$.

What are the types of roots a polynomial can have?

Roots can be real, complex, or repeated. A repeated root has a multiplicity greater than one, meaning its factor appears more than once in the factored form.

What's the difference between expanded form and factored form?

Expanded form writes a polynomial as a sum of separate terms, like $x^2 - x - 6$. Factored form writes the same polynomial as a product, like $(x-3)(x+2)$.