5.4 Dividing Polynomials

Dividing Polynomials

Dividing polynomials lets you break a complex polynomial into a simpler quotient and remainder, much like dividing integers. You'll use this skill constantly when factoring higher-degree polynomials, simplifying rational expressions, and solving equations. The two main techniques are long division and synthetic division.

Dividing Polynomials

Polynomial Long Division

Long division works for dividing by any polynomial divisor. The process mirrors numerical long division.

Steps for polynomial long division:

1. Write the dividend and divisor in standard form (descending powers). If any degree is missing, 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 as the first term of the quotient.
3. Multiply the entire divisor by that quotient term.
4. Subtract the result from the current dividend to get a new, smaller dividend.
5. Repeat steps 2–4 using the new dividend until the degree of what remains is less than the degree of the divisor.
6. Whatever is left over is the remainder.

Example: Divide $x^3 + 2x^2 - 5x - 6$ by $x + 3$.

• Divide $x^3 \div x = x^2$. Multiply: $x^2(x+3) = x^3 + 3x^2$. Subtract to get $-x^2 - 5x - 6$.
• Divide $-x^2 \div x = -x$. Multiply: $-x(x+3) = -x^2 - 3x$. Subtract to get $-2x - 6$.
• Divide $-2x \div x = -2$. Multiply: $-2(x+3) = -2x - 6$. Subtract to get $0$.

The quotient is $x^2 - x - 2$ with remainder $0$, so $x + 3$ divides evenly.

Synthetic Division

Synthetic division is a shortcut that works only when the divisor is of the form $x - c$ (a linear binomial with a leading coefficient of 1).

Steps for synthetic division:

1. Write the coefficients of the dividend in descending order of degree. Include 0 for any missing terms.

2. Write the value $c$ to the left. (If dividing by $x - 3$, use $3$. If dividing by $x + 3$, use $-3$.)

3. Bring the first coefficient straight down.

4. Multiply that number by $c$ and write the result under the next coefficient.

5. Add the column. Write the sum below the line.

6. Repeat steps 4–5 for each remaining coefficient.

7. The last number on the bottom row is the remainder. The other numbers are the coefficients of the quotient, which has degree one less than the dividend.

Example: Divide $x^3 + 2x^2 - 5x - 6$ by $x + 3$ (so $c = -3$).

• Coefficients: $1, \; 2, \; -5, \; -6$
• Bring down $1$. Multiply $1 \times (-3) = -3$; add to $2$ to get $-1$.
• Multiply $-1 \times (-3) = 3$; add to $-5$ to get $-2$.
• Multiply $-2 \times (-3) = 6$; add to $-6$ to get $0$.

Bottom row: $1, \; -1, \; -2, \; 0$. The quotient is $x^2 - x - 2$ with remainder $0$, matching the long division result.

The Remainder Theorem and Finding Factors

The Remainder Theorem says: when a polynomial $P(x)$ is divided by $x - c$, the remainder equals $P(c)$.

This gives you a quick test for factors. If $P(c) = 0$, then $x - c$ divides $P(x)$ evenly, meaning $x - c$ is a factor. This special case is called the Factor Theorem.

How to use it to find factors:

1. List candidate values of $c$. For a polynomial with integer coefficients, the Rational Root Theorem tells you to try $\pm$ (factors of the constant term) divided by $\pm$ (factors of the leading coefficient).

2. Evaluate $P(c)$ for each candidate (or use synthetic division).

3. Whenever $P(c) = 0$, you've found a factor $x - c$.

4. After dividing out that factor, repeat the process on the reduced quotient to find additional factors.

Example: Factor $x^3 - 2x^2 - 5x + 6$.

• Possible rational roots: $\pm 1, \pm 2, \pm 3, \pm 6$.
• $P(1) = 1 - 2 - 5 + 6 = 0$, so $x - 1$ is a factor.
• Divide by $x - 1$ (synthetic division with $c = 1$) to get $x^2 - x - 6$.
• Factor the quotient: $x^2 - x - 6 = (x - 3)(x + 2)$.
• Full factorization: $(x - 1)(x - 3)(x + 2)$.

Applications of Polynomial Division

Polynomial division shows up in geometry-style word problems where one quantity is expressed as a product of polynomials.

• Area problems: If a rectangle has area $x^3 + 2x^2 - 5x - 6$ and one side is $x + 3$, divide the area by that side to find the other side: $x^2 - x - 2$.
• Volume problems: If a solid has volume $3x^3 - 7x^2 + 5x - 1$ and the base area is $3x - 1$, divide volume by base area to find the height. (Note: since the divisor's leading coefficient isn't 1, you'd use long division here rather than synthetic division.)

Key Vocabulary

• Degree: The highest power of the variable in a polynomial (e.g., $x^3 + 2x$ has degree 3).
• Leading coefficient: The coefficient of the highest-degree term.
• Quotient: The result of the division (before the remainder).
• Remainder: What's left over after division. The full result is written as: $\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$.
• Rational expression: A fraction where both the numerator and denominator are polynomials. Polynomial division is often used to simplify these.