Remainder and factor theorem
Remainder and Factor Theorem
Remainder and Factor Theorem is a topic in Polynomial Functions in the California Common Core State Standards. It is aligned to Standard A-APR.2, which requires students to know and apply the Remainder Theorem: for a polynomial p(x), p(a) is the remainder on division by x - a, so p(a) = 0 means x - a is a factor.
The Remainder Theorem says \(P(c)\) is the remainder of \(P(x)\div(x-c)\), and the Factor Theorem says \((x-c)\) is a factor exactly when \(P(c)=0\).
Theory
Two linked theorems connect values and factors of \(P(x)\):
- Remainder Theorem: the remainder of \(P(x)\div(x-c)\) is \(P(c)\).
- Factor Theorem: \((x-c)\) is a factor \(\iff P(c)=0\).
The theorems:
How to use the theorems
- To find a remainder by \(x-c\), compute \(P(c)\).
- To test a factor \(x-c\), check whether \(P(c)=0\).
- If \(P(c)=0\), divide out \((x-c)\) to keep factoring.
- Solve for unknown coefficients using a known zero.
Evaluate \(P(2)\).
| \(P(2)\) | \(=\) | \(8-8+1\) |
| \(=\) | \(1\) |
Check whether \(P(3)=0\).
| \(P(3)\) | \(=\) | \(27-18-6-3\) |
| \(=\) | \(0\) |
Since \(P(3)=0\), \(x-3\) is a factor.
Set \(P(1)=0\).
| \(1+k-6\) | \(=\) | \(0\) |
| \(k\) | \(=\) | \(5\) |
\(P(2)=0\) means \((x-2)\) divides \(P\).
| \(x^3-3x^2+4\) | \(=\) | \((x-2)(x^2-x-2)\) |
| \(=\) | \((x-2)(x-2)(x+1)\) |
Common pitfalls
Frequently asked questions
What is the Remainder Theorem?
The remainder of \(P(x)\div(x-c)\) equals \(P(c)\).
What is the Factor Theorem?
\((x-c)\) is a factor of \(P(x)\) exactly when \(P(c)=0\).
How do you test if \(x-c\) is a factor?
Evaluate \(P(c)\); if it is \(0\), then \(x-c\) is a factor.
How are zeros and factors related?
Each zero \(x=c\) gives a factor \((x-c)\), and vice versa.