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Algebra 2
18. CC.HSA.APR.B.2
18.2 Remainder theorem
The Remainder Theorem
Dividing f(x) by (x-a) leaves a remainder equal to f(a) -- no division required.
What You'll Learn
Dividing f(x) by (x-a) leaves a remainder equal to f(a) -- just evaluate, don't divide.
This works because f(x) = (x-a)*q(x) + r, and substituting x=a makes the first term 0.
Example: f(x)=x^3-2x^2+3x-5 divided by (x-2) has remainder f(2)=1.
Synthetic division confirms the same remainder without needing the theorem.
When f(a)=0, the remainder is 0 and (x-a) is a factor -- the factor theorem.
What You'll Practice
1
Finding remainders using the Remainder Theorem for divisors like (2x - 5)
2
Comparing synthetic division results with Remainder Theorem calculations
3
Solving for unknown coefficients using given remainder conditions
4
Working with polynomials containing multiple variables and missing terms
Why This Matters
The Remainder Theorem gives you a powerful shortcut for polynomial division that saves time on tests and homework. Instead of lengthy synthetic or long division, you simply plug in a value and evaluatea skill essential for factoring, graphing, and advanced algebra topics.
Before You Start — Make Sure You Can:
This Unit Includes
4 Video lessons
Practice exercises
Learning resources
Skills
Remainder Theorem
Polynomials
Synthetic Division
Polynomial Division
Evaluation
Systems of Equations
IL Curriculum Aligned