Teacher Guide: Synthetic Division

Learn a faster method for dividing polynomials by linear factors using synthetic division.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Polynomials. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Perform synthetic division to divide polynomials by linear binomials
Set up synthetic division correctly, including zeros for missing terms
Interpret the result as quotient and remainder
Apply the Remainder Theorem to evaluate polynomials
Use synthetic division to test potential roots of polynomials

Prerequisites

• Understanding of polynomial terminology (degree, coefficient, terms)
• Polynomial long division
• Adding, subtracting, and multiplying integers
• Evaluating polynomials at specific values

Discussion Starters

1. Why is synthetic division faster than polynomial long division?
2. How can you tell if is a factor just from the synthetic division result?
3. What happens if you try synthetic division with a divisor like ? Why doesn't it work directly?
4. How does the Remainder Theorem let you evaluate polynomials without substituting?

Common Misconceptions

Thinking synthetic division works for any polynomial divisor

Confusing which sign to use for

Forgetting that the quotient's degree is one less than the dividend's

Differentiation Ideas

For Struggling Students:

• Use a synthetic division template with labeled columns
• Practice with polynomials that have all terms present first
• Color-code the multiply and add steps

For On-Level Students:

• Include problems with missing terms
• Practice with both and divisors
• Apply Remainder Theorem to evaluate polynomials

For Advanced Students:

• Use synthetic division to completely factor polynomials
• Explore connection to Horner's method for efficient evaluation
• Investigate what modifications allow division by

Standards Alignment

A-APR.D.6 (CCSS.MATH.CONTENT.HSA.APR.D.6)
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x)
A-APR.B.2 (CCSS.MATH.CONTENT.HSA.APR.B.2)
Know and apply the Remainder Theorem

Lesson Resources

visualStep-by-Step Synthetic Division Tool
Interactive tool showing each step of the algorithm
activityRoot Testing Challenge
Use synthetic division to find roots of polynomials
worksheetPractice Problems
Graduated difficulty from basic to complex divisions

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form .

Instead of using long division, we work only with the coefficients and perform simple arithmetic operations.

When to use synthetic division:

The divisor must be linear: or
The coefficient of in the divisor must be

The setup: To divide by : 1. Write the value of (the root) on the left 2. Write the coefficients of in order 3. Include for any missing terms

Worked Examples

Divide by

Identify the divisor value

means →

Write coefficients in order

has coefficients: →

Set up the synthetic division

→ Ready to calculate

Bring down the first coefficient

→ First coefficient:

Multiply and add for each column

, then , then , then →

Write the quotient and remainder

Bottom row: are quotient coefficients; is remainder → Quotient: , Remainder:

Answer: Since the remainder is , is a factor of the polynomial.

Common Mistakes

Forgetting to include zeros for missing terms

Why it's wrong: Every power of from highest to constant must have a coefficient. Missing terms have coefficient .

Correct: For , write: (zeros for , , and )

Using the wrong sign for c when dividing by

Why it's wrong: Synthetic division uses the root form . When dividing by , you need .

Correct: , so use in the synthetic division setup

Adding when you should multiply, or vice versa

Why it's wrong: The pattern is: multiply by , then add to the next coefficient.

Correct: Always: bring down first coefficient, then repeat (multiply by , add to next coefficient)

Writing the quotient with the wrong degree

Why it's wrong: The quotient has one degree less than the original polynomial.

Correct: If dividing a cubic () by linear, the quotient is quadratic ()

Why It Matters

Synthetic division is a powerful tool that saves time and reduces errors:

Speed: Much faster than polynomial long division
Simplicity: Only uses basic arithmetic (addition and multiplication)
Factoring: Helps find roots of polynomials quickly
Remainder Theorem: If you divide by , the remainder equals

Real applications:

Engineering: Analyzing transfer functions in control systems
Computer graphics: Polynomial curve calculations
Finance: Modeling growth patterns with polynomial functions

Real World Applications

Engineering: Transfer Functions

Engineers use polynomial division when analyzing control systems. The transfer function of a system is often a ratio of polynomials.

Example:

If a system has transfer function , synthetic division simplifies it to .

1Try It Yourself

A filter circuit has polynomial response . You need to factor it by testing if is a root.

Use synthetic division to test if is a factor.

Step 1: Write the mathematical expression

Set up: coefficients are with

Computer Science: Algorithm Optimization

Evaluating polynomials efficiently uses ideas from synthetic division. Horner's method is essentially synthetic division for function evaluation.

Example:

To find for , use synthetic division with . The remainder equals .

2Try It Yourself

You need to evaluate at for a graphics calculation.

Use synthetic division to find .

Step 1: Write the mathematical expression

Coefficients: (remember the missing term!)

Key Takeaways

1Synthetic division is a shortcut for dividing polynomials by
2Use only the coefficients, including for missing terms
3Pattern: bring down, multiply by , add, repeat
4The last number is the remainder; other numbers are quotient coefficients
5For , use in the setup
6Remainder Theorem: the remainder equals

Frequently Asked Questions

Can I use synthetic division for any divisor?

No, synthetic division only works when dividing by a linear binomial where the coefficient of is . For other divisors like or , use polynomial long division.

Why do I use when dividing by ?

Synthetic division is based on the root form . Since , the value of is . Think of it as finding what makes the divisor equal zero: means .

How does the Remainder Theorem connect to synthetic division?

When you divide by , the remainder you get equals - the value of the polynomial at . This means you can use synthetic division to quickly evaluate polynomials!

Glossary

Synthetic division: A shortcut method for dividing a polynomial by a linear binomial using only coefficients
Divisor: The polynomial you are dividing by; in synthetic division, it must be of the form
Quotient: The result of the division (excluding the remainder)
Remainder: The amount left over after division; equals by the Remainder Theorem
Remainder Theorem: States that when is divided by , the remainder equals
Factor Theorem: States that is a factor of if and only if

Formula Card

Synthetic Division Setup

Write $c$ on left, coefficients in descending order on right

The Algorithm

Bring down Multiply by Add Repeat

Continue until all coefficients are processed

Result Interpretation

Last number is remainder $R$; others form quotient $Q(x)$

Remainder Theorem

The remainder equals the polynomial evaluated at $c$