Teacher Guide: Dividing Polynomials
Master polynomial division using long division, synthetic division, and factoring techniques.
Use this lesson with your class
Free, no student accounts needed.
Share with students
Students open the lesson and practise with instant feedback.
Class quiz
10 questions on Polynomials. Students join with a name, you see everyone's score.
For Teachers
- Divide polynomials by monomials by dividing each term
- Perform polynomial long division with linear and quadratic divisors
- Apply synthetic division to divide by linear factors of the form
- Use the Remainder Theorem to evaluate polynomials
- Apply the Factor Theorem to determine factors of polynomials
- • Multiplying polynomials and the FOIL method
- • Factoring polynomials, including difference of squares
- • Understanding of numerical long division
- • Basic exponent rules
- 1. Why does synthetic division only work when the divisor has a leading coefficient of 1?
- 2. How is polynomial long division similar to the long division you learned with numbers?
- 3. If , what can you conclude about and the polynomial ?
- 4. Can you think of a situation where knowing the remainder is more useful than finding the quotient?
Synthetic division works for any divisor
The remainder must always be a number
For Struggling Students:
- • Start with monomial division only before introducing long division
- • Provide pre-made synthetic division templates with boxes to fill in
- • Use numerical long division review to connect to polynomial division
For On-Level Students:
- • Practice all three methods with increasing complexity
- • Apply the Remainder Theorem to check answers
- • Solve problems requiring method selection
For Advanced Students:
- • Extend synthetic division to divisors like with coefficient adjustments
- • Explore partial fraction decomposition
- • Apply polynomial division to solve cubic and quartic equations
- HSA-APR.D.6 (CCSS.MATH.CONTENT.HSA.APR.D.6)
Rewrite simple rational expressions in different forms using polynomial long division
- HSA-APR.B.2 (CCSS.MATH.CONTENT.HSA.APR.B.2)
Know and apply the Remainder Theorem
- visualStep-by-Step Division Animator
Watch polynomial long division unfold step by step
- activityDivision Method Chooser
Practice selecting the best division method for each problem
- worksheetDivision Practice Set
Mixed problems with monomial, long, and synthetic division
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
Worked Examples
Divide:
Set up the division
Divide each term of the numerator by →
Divide the first term
→
Divide the second term
→
Divide the third term
→
Combine results
Put all terms together →
Answer:
Common Mistakes
Forgetting to include zero coefficients for missing terms
Why it's wrong: In , the and terms have coefficient and must be included in synthetic division.
Correct: Write before dividing
Using the wrong sign in synthetic division
Why it's wrong: For , use , not . The sign is opposite to what appears in the divisor.
Correct: means use ; means use
Incorrect subtraction in long division
Why it's wrong: Students often add instead of subtract when eliminating terms.
Correct: Always subtract the entire product: change all signs before adding
Why It Matters
- Simplifying expressions: Reduce complex fractions to simpler forms
- Finding roots: Factor higher-degree polynomials to find zeros
- Calculus preparation: Integration by partial fractions requires polynomial division
- Engineering: Signal processing and control systems use polynomial operations
Real World Applications
Engineering: Transfer Functions
Control engineers divide polynomials to analyze system behavior and design stable controllers.
Example:
A system's transfer function simplifies to after polynomial division.
An engineer needs to simplify for circuit analysis.
What is the simplified form?
Step 1: Write the mathematical expression
Factor and simplify:
Computer Graphics: Bezier Curves
Polynomial division helps split curves for rendering and animation in video games and CGI.
Example:
Dividing a cubic Bezier curve into smaller segments requires polynomial division at specific parameter values.
A graphics programmer needs to find if is a factor of .
Is a factor? Find the quotient.
Step 1: Write the mathematical expression
Use synthetic division with :
Key Takeaways
- 1To divide by a monomial, divide each term of the polynomial separately
- 2Polynomial long division follows the same process as numerical long division
- 3Synthetic division is a shortcut for dividing by : use in the process
- 4Always include zero coefficients for missing terms in the dividend
- 5The Remainder Theorem: dividing by gives remainder
Frequently Asked Questions
When should I use synthetic division vs. long division?
How do I check if my polynomial division is correct?
What if the remainder is not zero?
Glossary
- Dividend
- The polynomial being divided (the numerator)
- Divisor
- The polynomial we are dividing by (the denominator)
- Quotient
- The result of the division (without the remainder)
- Remainder
- What is left over after division that cannot be divided further
- Synthetic division
- A shortcut method for dividing a polynomial by a linear factor
Formula Card
Division Algorithm
Relates dividend, divisor, quotient, and remainder
Remainder Theorem
Evaluate P(c) to find the remainder when dividing by (x - c)
Factor Theorem
If P(c) = 0, then (x - c) divides P(x) evenly