What is Polynomial Division and Remainder Theorem?

Understanding polynomial long division, synthetic division, and the Remainder and Factor Theorems

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Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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What course covers Polynomial Division and Remainder Theorem?

Polynomial Division and Remainder Theorem is part of the AP Precalculus course on Study Mondo, specifically in the Polynomial and Rational Functions section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Polynomial Division and Remainder Theorem?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Polynomial Division and Remainder Theorem

Why Divide Polynomials?

❓ Question:

Use synthetic division to divide $f(x) = 2x^3 - 5x^2 + x + 2$ by $(x - 2)$ .

Solution:

Step 1: Set up synthetic division with $c = 2$ .

The coefficients of $f(x)$ are: $2, -5, 1, 2$

❓ Question:

Use the Remainder Theorem to find the remainder when $f(x) = x^4 - 3x^3 + 2x - 5$ is divided by .

❓ Question:

Find all rational zeros of $f(x) = 2x^3 - x^2 - 13x - 6$ and factor completely.

❓ Question:

Use synthetic division to divide: (2x³ - 5x² + 3x - 7) ÷ (x - 2)

Step 1: Set up synthetic division with divisor x - 2 (use 2): 2 | 2 -5 3 -7

Step 2: Bring down the first coefficient: 2 | 2 -5 3 -7 ↓ 2

Step 3: Multiply and add: 2 | 2 -5 3 -7 4 -2 2 ――――――――――――― 2 -1 1 -5

Step 4: Write the result: Quotient: 2x² - x + 1 Remainder: -5

Step 5: Express as quotient + remainder/divisor: (2x³ - 5x² + 3x - 7) = (x - 2)(2x² - x + 1) - 5

Answer: Quotient: 2x² - x + 1, Remainder: -5

❓ Question:

Use the Remainder Theorem to find the remainder when f(x) = x⁴ - 3x³ + 2x - 5 is divided by (x + 1).

Step 1: Recall the Remainder Theorem: When f(x) is divided by (x - c), the remainder is f(c)

Step 2: Identify c from (x + 1): x + 1 = x - (-1) So c = -1

Step 3: Evaluate f(-1): f(-1) = (-1)⁴ - 3(-1)³ + 2(-1) - 5 = 1 - 3(-1) - 2 - 5 = 1 + 3 - 2 - 5 = -3

Step 4: Conclusion: By the Remainder Theorem, the remainder is -3

Step 5: Verify with synthetic division if desired: -1 | 1 -3 0 2 -5 -1 4 -4 2 ―――――――――――――――――― 1 -4 4 -2 -3 ✓

Answer: -3

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

Solution:

The Remainder Theorem states that the remainder when dividing $f(x)$ by $(x - c)$ is $f(c)$ .

Step 1: Identify $c$ .

We're dividing by $(x + 1) = (x - (-1))$ , so $c = -1$ .

Step 2: Evaluate $f(-1)$ .

$f(-1) = (-1)^4 - 3(-1)^3 + 2(-1) - 5$

$= 1 - 3(-1) - 2 - 5$

$= 1 + 3 - 2 - 5$

$= -3$

Answer: The remainder is $-3$ .

Note: We found this without doing any division!

Solution:

Step 1: List possible rational zeros using the Rational Zero Theorem.

Factors of constant term (-6): $\pm 1, \pm 2, \pm 3, \pm 6$ Factors of leading coefficient (2): $\pm 1, \pm 2$

Possible rational zeros: $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}$

Step 2: Test candidates.

Try $x = -1$ : $f(-1) = 2(-1)^3 - (-1)^2 - 13(-1) - 6$ ✗

Try $x = 2$ : $f(2) = 2(2)^3 - (2)^2 - 13(2) - 6$ ✗

Try $x = -2$ : $f(-2) = 2(-2)^3 - (-2)^2 - 13(-2) - 6$ ✓

Step 3: Use synthetic division with $x = -2$ .

-2 & 2 & -1 & -13 & -6 \\ & & -4 & 10 & 6 \\ \hline & 2 & -5 & -3 & 0 \end{array}$$ Quotient: $2x^2 - 5x - 3$ Step 4: Factor the quotient. $$2x^2 - 5x - 3 = (2x + 1)(x - 3)$$ Step 5: Write complete factorization. $$f(x) = (x + 2)(2x + 1)(x - 3)$$ **Zeros**: $x = -2, x = -\frac{1}{2}, x = 3$ **Answer:** Zeros are $-2, -\frac{1}{2}, 3$; Factored form: $(x + 2)(2x + 1)(x - 3)$

Polynomial Division and Remainder Theorem

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Polynomial Division and Remainder Theorem

Why Divide Polynomials?

Long Division of Polynomials

Steps for Polynomial Long Division

Example Format

Synthetic Division

When to Use Synthetic Division

Steps for Synthetic Division

Example Setup

📚 Practice Problems

1Problem 1medium

❓ Question:

2Problem 2easy

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3Problem 3hard

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4Problem 4medium

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5Problem 5hard

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⚠️ Common Mistakes: Polynomial Division and Remainder Theorem

🌍 Real-World Applications: Polynomial Division and Remainder Theorem

📝 Worked Example: Related Rates — Expanding Circle

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