Solved Math Exams

What is Polynomial Long Division?

You already know how to divide numbers using long division. Polynomial long division is the exact same process — just with polynomials instead of digits.

Why do we need this? When you have a rational expression like $\frac{x^3 + 2x^2 - 5x + 1}{x - 2}$, you can't always simplify it by factoring. Polynomial long division lets you divide these expressions to get a simpler form — often a polynomial plus a remainder.

This technique is essential for:

Simplifying complex rational expressions
Finding oblique (slant) asymptotes
Verifying factors of polynomials

The Process: Divide, Multiply, Subtract, Repeat

Polynomial long division follows the same steps as numerical long division:

Divide: Divide the leading term of the dividend by the leading term of the divisor
Multiply: Multiply the entire divisor by that result
Subtract: Subtract to get a new polynomial
Repeat: Use the result as your new dividend and repeat until the degree is less than the divisor

Setting Up the Problem

When dividing $\frac{x^3 + 2x^2 - 5x + 1}{x - 2}$:

Dividend (inside): $x^3 + 2x^2 - 5x + 1$
Divisor (outside): $x - 2$
Quotient (answer on top): what we're finding
Remainder: what's left over

⚠️ Important: Make sure both polynomials are written in descending order of powers. If any powers are missing, include them with coefficient 0 (e.g., $x^3 + 1$ becomes $x^3 + 0x^2 + 0x + 1$).

Example 1: Basic Polynomial Division

Divide $\frac{x^3 + 2x^2 - 5x + 1}{x - 2}$.

Step 1: Divide the leading terms

Divide $x^3 \div x = x^2$. Write $x^2$ on top. Then multiply: $x^2(x-2) = x^3 - 2x^2$. Subtract this from the dividend.

              x²
            ___________________
    x - 2  ) x³  + 2x² - 5x + 1
             x³  - 2x²
             ──────────
                  4x²

We get $4x^2$. Bring down the $-5x$.

Step 2: Repeat with the new leading term

Divide $4x^2 \div x = 4x$. Write $+4x$ on top. Multiply: $4x(x-2) = 4x^2 - 8x$. Subtract.

              x²  + 4x
            ___________________
    x - 2  ) x³  + 2x² - 5x + 1
             x³  - 2x²
             ──────────
                  4x² - 5x
                  4x² - 8x
                  ─────────
                       3x

We get $3x$. Bring down the $+1$.

Step 3: One more round

Divide $3x \div x = 3$. Write $+3$ on top. Multiply: $3(x-2) = 3x - 6$. Subtract.

              x²  + 4x  + 3
            ___________________
    x - 2  ) x³  + 2x² - 5x + 1
             x³  - 2x²
             ──────────
                  4x² - 5x
                  4x² - 8x
                  ─────────
                       3x + 1
                       3x - 6
                       ──────
                            7

The remainder is $7$. Since $7$ has degree 0 (less than the divisor's degree 1), we stop.

Final Answer:

$$\boxed{\frac{x^3 + 2x^2 - 5x + 1}{x - 2} = x^2 + 4x + 3 + \frac{7}{x - 2}}$$

Example 2: Missing Terms

Divide $\frac{x^3 - 8}{x - 2}$.

Step 1: Fill in missing terms and set up

The dividend is missing $x^2$ and $x$ terms. Rewrite as $x^3 + 0x^2 + 0x - 8$.

Divide $x^3 \div x = x^2$. Multiply and subtract.

              x²
            ___________________
    x - 2  ) x³  + 0x² + 0x - 8
             x³  - 2x²
             ──────────
                  2x²

We get $2x^2$. Bring down the $+0x$.

Step 2: Continue

Divide $2x^2 \div x = 2x$. Multiply and subtract.

              x²  + 2x
            ___________________
    x - 2  ) x³  + 0x² + 0x - 8
             x³  - 2x²
             ──────────
                  2x² + 0x
                  2x² - 4x
                  ─────────
                       4x

We get $4x$. Bring down the $-8$.

Step 3: Final round

Divide $4x \div x = 4$. Multiply and subtract.

              x²  + 2x  + 4
            ___________________
    x - 2  ) x³  + 0x² + 0x - 8
             x³  - 2x²
             ──────────
                  2x² + 0x
                  2x² - 4x
                  ─────────
                       4x - 8
                       4x - 8
                       ──────
                            0

The remainder is $0$ — it divides evenly!

Final Answer:

$$\boxed{\frac{x^3 - 8}{x - 2} = x^2 + 2x + 4}$$

💡 This confirms that $(x - 2)$ is a factor of $x^3 - 8$, and $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$.

Writing the Final Answer

Your answer can be written in two equivalent forms:

Form 1: Quotient + Fraction $$\frac{\text{dividend}}{\text{divisor}} = \text{quotient} + \frac{\text{remainder}}{\text{divisor}}$$

Form 2: Multiplication $$\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$$

Common Mistakes and Misunderstandings

❌ Mistake: Forgetting to include missing terms

Wrong: Dividing $x^3 - 1$ by $x - 1$ without placeholders

Why it's wrong: When you subtract, the columns won't align properly, leading to errors.

Correct: Rewrite as $x^3 + 0x^2 + 0x - 1$ before dividing.

❌ Mistake: Subtracting incorrectly (sign errors)

Wrong: $(4x^2 - 5x) - (4x^2 - 8x) = -3x$

Why it's wrong: When subtracting, you must distribute the negative: $-5x - (-8x) = -5x + 8x = 3x$

Correct: $(4x^2 - 5x) - (4x^2 - 8x) = 3x$

❌ Mistake: Stopping too early or too late

Wrong: Continuing to divide when the remainder's degree is already less than the divisor's degree.

Why it's wrong: You stop when you can't divide anymore — when the remainder's degree is less than the divisor's degree.

Correct: If dividing by $(x - 2)$ (degree 1) and you get remainder $7$ (degree 0), you're done!

Polynomial Long Division for MATH 140

Exam Relevance for MATH 140