Lesson 6: Divide Polynomials

New Concept

This lesson introduces two key methods for polynomial division. You'll learn how to divide a polynomial by a monomial using fraction properties, and master the long division process for dividing a polynomial by a binomial.

What’s next

Now that you have the big picture, get ready for worked examples and interactive practice cards that break down each division method step-by-step.

Property

Fraction Addition
If $a, b,$ and $c$ are numbers where $c \neq 0$, then $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$. We use this form to divide a polynomial by a monomial.

Division of a Polynomial by a Monomial
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Examples

• To find the quotient of $(21x^3 - 49x^2) \div 7x$, rewrite it as $\frac{21x^3}{7x} - \frac{49x^2}{7x}$, which simplifies to $3x^2 - 7x$.
• The quotient for $\frac{15a^4+25a^2}{-5a^2}$ is found by separating terms: $\frac{15a^4}{-5a^2} + \frac{25a^2}{-5a^2}$, resulting in $-3a^2 - 5$.
• For $\frac{18x^2y^3 + 27x^3y^2}{9x^2y}$, we divide each term to get $\frac{18x^2y^3}{9x^2y} + \frac{27x^3y^2}{9x^2y}$, which simplifies to $2y^2 + 3xy$.

Property

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers.

1. Write the long division with terms in descending order of degree.
2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
3. Multiply the result by the entire divisor and subtract it from the dividend.
4. Bring down the next term and repeat the process.

Examples

• In the division of $(x^2 + 8x + 15)$ by $(x+3)$, the process of polynomial long division yields a final quotient of $x+5$.
• For $(2x^2 + x - 15) \div (x+3)$, the first step is dividing $2x^2$ by $x$ to get $2x$. The process continues until you find the full quotient is $2x-5$.
• When dividing $(a^2 - 10a + 21)$ by $(a-7)$, long division yields a quotient of $a-3$ with no remainder.

Explanation

This is a step-by-step process for dividing complex polynomials. By repeatedly focusing on the leading terms, you systematically find the quotient, just like you would with large numbers. It organizes a complex problem into simple steps.

Property

When polynomial long division does not result in zero, the final leftover polynomial is the remainder. The final answer is written as the quotient plus a fraction where the remainder is the numerator and the divisor is the denominator.

Quotient + $\frac{\text{Remainder}}{\text{Divisor}}$

Examples

• Dividing $(x^2 + 5x + 7)$ by $(x+2)$ leaves a remainder of $1$, so the answer is written as the quotient plus a fraction: $x+3 + \frac{1}{x+2}$.
• When dividing $(2y^2 - 3y + 5)$ by $(y-1)$, the quotient is $2y-1$ and the remainder is $4$. This is written as $2y-1 + \frac{4}{y-1}$.
• For $(a^3 - a^2 + a + 5) \div (a+1)$, the quotient is $a^2-2a+3$ with a remainder of $2$, so the full answer is $a^2-2a+3 + \frac{2}{a+1}$.

Property

When a dividend is missing a term for a specific degree, you must add in a placeholder with a coefficient of 0. For example, to divide $x^4 - x^2 + 5x - 2$, it is missing an $x^3$ term, so we add in $0x^3$ as a placeholder, writing it as $x^4 + 0x^3 - x^2 + 5x - 2$.

Examples

• To divide $x^3 - 27$ by $x-3$, first rewrite the dividend with placeholders for the missing $x^2$ and $x$ terms as $x^3 + 0x^2 + 0x - 27$.
• For the problem $(y^4 - 5y^2 + 10) \div (y+2)$, set up the long division using $y^4 + 0y^3 - 5y^2 + 0y + 10$ as the dividend.
• To divide $2a^3 + 3a - 1$ by $2a+1$, the dividend must be written as $2a^3 + 0a^2 + 3a - 1$ to keep the columns properly aligned.

Explanation

Placeholders are essential for keeping your work organized. They hold the spot for a missing power of the variable, ensuring that you correctly align like terms when you subtract during long division, just like writing 502 instead of 52.