Dividing Polynomials

Dividing monomials

We are now familiar with all the properties of exponents and have used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

Example. Find the quotient: $54a^2b^3 \div \left(-6ab^5\right)$.

When we divide monomials with more than one variable, we write one fraction for each variable:

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example. Find the quotient: $\tfrac{14x^7y^{12}}{21x^{11}y^6}$.

Be very careful to simplify $\tfrac{14}{21}$ by dividing out a common factor, and to simplify the variables by subtracting their exponents:

Divide a polynomial by a monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.

The method is based on the properties of fraction addition. The sum $\tfrac{y}{5}+\tfrac{2}{5}$ simplifies to $\tfrac{y+2}{5}$. In reverse, $\tfrac{y+2}{5}$ can be split into $\tfrac{y}{5}+\tfrac{2}{5}$. More generally, if $a$, $b$, and $c$ are numbers where $c\neq0$, then

Example. Find the quotient: $\left(18x^3y-36xy^2\right)\div(-3xy)$.

Divide polynomials using long division

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. Consider dividing $875$ by $25$:

We check division by multiplying the quotient by the divisor. If we did the division correctly, the product should equal the dividend: $35\cdot25=875$ ✓.

Example. Find the quotient: $\left(x^2+9x+20\right)\div(x+5)$.

Write the dividend in standard form. Divide $x^2$ by $x$ to get $x$ and put it in the quotient. Multiply $x$ by $x+5$, subtract, and bring down $20$. Then divide $4x$ by $x$ to get $4$, multiply, and subtract:

To check, multiply $(x+4)(x+5)$. The product is $x^2+9x+20$ ✓, so the quotient is $x+4$.

Sometimes polynomial division leaves a remainder. We write the remainder as a fraction with the divisor as the denominator. When a degree is missing from the dividend, we insert a term with zero coefficient as a placeholder.

Example. Find the quotient: $\left(x^4-x^2+5x-6\right)\div(x+2)$.

There is no $x^3$ term, so add $0x^3$ as a placeholder. Long division gives

The quotient is

To check, multiply $(x+2)\left(x^3-2x^2+3x-1-\tfrac{4}{x+2}\right)$. The result is $x^4-x^2+5x-6$.

In the next example, we divide by $2a+3$. We must consider the constants as well as the variables.

Example. Find the quotient: $\left(8a^3+27\right)\div(2a+3)$.

Add the two placeholders $0a^2$ and $0a$ before dividing:

To check, multiply $(2a+3)(4a^2-6a+9)$. The product is $8a^3+27$, so the quotient is $4a^2-6a+9$.

Divide polynomials using synthetic division

Long division can be tedious, so mathematicians use a pattern called synthetic division. It removes repeated variables and numbers from the long-division work. The first row contains the coefficients of the dividend; the second row contains the successive products; and the third row contains the coefficients of the quotient followed by the remainder.

Synthetic division only works when the divisor is of the form $x-c$.

Example. Use synthetic division to find the quotient and remainder when $2x^3+3x^2+x+8$ is divided by $x+2$.

Write the divisor as $x-(-2)$, so $c=-2$. Write the coefficients of the dividend in the first row. Bring down the first coefficient. Then repeatedly multiply by $-2$ and add the next column:

The numbers $2,-1,3$ are the coefficients of the quotient, and the last number is the remainder. The quotient is $2x^2-x+3$ and the remainder is $2$.

Check:

In the next example, we do all the steps together.

Example. Use synthetic division to find the quotient and remainder when $x^4-16x^2+3x+12$ is divided by $x+4$.

The polynomial is written in descending degree, but there is no $x^3$ term, so use $0$ as a placeholder. Since $x+4=x-(-4)$, use $c=-4$:

We divided a fourth-degree polynomial by a first-degree polynomial, so the quotient is third degree. Reading from the last row, the quotient is $x^3-4x^2+3$, and the remainder is $0$.

Divide polynomial functions

Just as polynomials can be divided, polynomial functions can also be divided.

Example. For functions $f(x)=x^2-5x-14$ and $g(x)=x+2$, find (a) $\left(\tfrac{f}{g}\right)(x)$ and (b) $\left(\tfrac{f}{g}\right)(-4)$.

For part (a), substitute the function rules and divide the polynomials:

For part (b), substitute $x=-4$ into the quotient found in part (a):

Use the Remainder and Factor Theorems

Look at division problems that end with a remainder. When the divisor is written as $x-c$, the value $f(c)$ is the same as the remainder from the division.

To see this generally, a division problem can be checked by multiplying the quotient $q(x)$ by the divisor $x-c$ and adding the remainder $r$:

Evaluating at $c$ gives

Example. Use the Remainder Theorem to find the remainder when $f(x)=x^3+3x+19$ is divided by $x+2$.

Write $x+2$ as $x-(-2)$, so $c=-2$. Evaluate $f(-2)$:

The remainder is $5$.

When we divided $8a^3+27$ by $2a+3$, the quotient was $4a^2-6a+9$ and the remainder was zero. Thus

so both $4a^2-6a+9$ and $2a+3$ are factors of $8a^3+27$.

Example. Use the Factor Theorem to determine if $x-4$ is a factor of $f(x)=x^3-64$.

The Factor Theorem tells us that $x-4$ is a factor if $f(4)=0$:

Since $f(4)=0$, $x-4$ is a factor of $f(x)=x^3-64$.

Key terms

division of a polynomial by a monomial — dividing each term of the polynomial by the monomial. polynomial long division — a procedure for dividing a polynomial by a binomial: divide, multiply, subtract, and bring down, repeating until the remainder has lower degree than the divisor. placeholder — a term with zero coefficient inserted for a missing degree. synthetic division — a shortened form of polynomial division using only coefficients, for a divisor of the form $x-c$. Remainder Theorem — when $f(x)$ is divided by $x-c$, the remainder is $f(c)$. Factor Theorem — $x-c$ is a factor of $f(x)$ exactly when $f(c)=0$.

This section is adapted from Intermediate Algebra 2e, Section 5.4: Dividing Polynomials by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed the multi-row worked-example step tables into prose and typeset math, recreated the numeric and polynomial long divisions and synthetic-division layouts as math arrays, omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises, and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.