5.4 Dividing Polynomials

Division of Polynomials

Dividing polynomials extends the long division you already know from arithmetic into the world of variables and exponents. It's a core technique you'll rely on to simplify rational expressions, find roots of polynomials, and factor higher-degree expressions.

Division of Monomials

Dividing one monomial by another comes down to two moves: divide the coefficients and subtract the exponents of like bases.

• $\frac{12x^5}{3x^2} = 4x^3$ because $12 \div 3 = 4$ and $x^{5-2} = x^3$

If subtracting exponents gives you a negative result, flip the variable to the denominator so the exponent becomes positive.

• $\frac{3x^2}{6x^5} = \frac{1}{2x^3}$ because $3 \div 6 = \frac{1}{2}$ and $x^{2-5} = x^{-3} = \frac{1}{x^3}$

Polynomials Divided by Monomials

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial separately.

• $\frac{6x^3 + 9x^2 - 12x}{3x} = \frac{6x^3}{3x} + \frac{9x^2}{3x} - \frac{12x}{3x} = 2x^2 + 3x - 4$

If a term in the numerator isn't evenly divisible by the monomial, it stays as a fraction in the result.

• $\frac{4x^3 + 6x^2 + 5}{2x} = 2x^2 + 3x + \frac{5}{2x}$

That leftover fraction ($\frac{5}{2x}$) makes the result a rational expression rather than a polynomial.

Long Division for Polynomials

Polynomial long division works just like numerical long division. Use it when the divisor has two or more terms.

1. Arrange both the dividend and divisor in descending order of degree. If any degree is missing (e.g., no $x^2$ term), insert a placeholder with a coefficient of 0.
2. Divide the leading term of the dividend by the leading term of the divisor. Write the result above the division bar.
3. Multiply that result by the entire divisor.
4. Subtract the product from the current dividend. (Watch your signs here; sign errors are the most common mistake.)
5. Bring down the next term and repeat steps 2–4.
6. Stop when the degree of what's left (the remainder) is less than the degree of the divisor.

The answer is written as:

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

For example, dividing $2x^3 + 3x^2 - 5x + 1$ by $x + 2$: you'd first divide $2x^3$ by $x$ to get $2x^2$, multiply $2x^2(x+2) = 2x^3 + 4x^2$, subtract, bring down the next term, and continue.

Synthetic Division of Polynomials

Synthetic division is a shortcut that only works when you're dividing by a linear expression of the form $x - a$.

1. Write the coefficients of the dividend in descending order. Include 0 for any missing terms. To the left, write the value $a$ (the number that makes the divisor equal zero). For $x - 3$, use $3$. For $x + 4$, use $-4$.

2. Bring the leading coefficient straight down below the line.

3. Multiply that number by $a$ and write the product under the next coefficient.

4. Add the column (next coefficient + product) and write the sum below the line.

5. Repeat steps 3–4 for every remaining coefficient.

6. The final number below the line is the remainder. All the other numbers are the coefficients of the quotient, which has degree one less than the dividend.

Remainder and Factor Theorems

These two theorems connect polynomial division to evaluating and factoring polynomials.

Remainder Theorem: When you divide a polynomial $P(x)$ by $x - a$, the remainder equals $P(a)$.

This means you can find the remainder without doing the full division. Just plug $a$ into the polynomial. For instance, if $P(x) = x^3 - 4x + 2$ and you divide by $x - 1$, the remainder is $P(1) = 1 - 4 + 2 = -1$.

Factor Theorem: $x - a$ is a factor of $P(x)$ if and only if $P(a) = 0$.

This is a direct consequence of the Remainder Theorem. If the remainder is zero, the divisor goes in evenly, which means it's a factor. So to test whether $x - 3$ is a factor of some polynomial, just check whether $P(3) = 0$.

Together, these theorems give you a fast way to find roots and factors of polynomials without fully factoring or graphing.