The Remainder Theorem and Factor Theorem connect polynomial division to simple substitution. Instead of doing full long division every time, you can evaluate a polynomial P(x) at x = a to learn the remainder when dividing by x - a. This matters because it makes checking factors, roots, and division results much faster.
These theorems are central tools in algebra, precalculus, and graphing polynomial functions.
When a polynomial P(x) is divided by x - a, the remainder is exactly P(a). If P(a) = 0, then the remainder is zero, so x - a divides evenly into P(x). That means x - a is a factor of the polynomial, and a is a root or zero of the function.
This creates a powerful bridge between algebraic factoring, polynomial graphs, and solving equations.
Key Facts
- Remainder Theorem: When P(x) is divided by x - a, the remainder is P(a).
- Factor Theorem: x - a is a factor of P(x) if and only if P(a) = 0.
- If P(a) = 0, then a is a root, zero, and x-intercept of y = P(x), if the graph is real-valued.
- Polynomial division form: P(x) = (x - a)Q(x) + R, where R = P(a).
- To test whether x - a is a factor, substitute a into P(x), not -a.
- For a divisor x + b, rewrite it as x - (-b), so the test value is x = -b.
Vocabulary
- Polynomial
- An expression made from constants, variables, and nonnegative integer powers, such as 3x^3 - 2x + 5.
- Remainder
- The amount left over after dividing one polynomial by another polynomial.
- Factor
- A polynomial that divides another polynomial evenly with a remainder of zero.
- Root
- A value of x that makes a polynomial equal to zero.
- Synthetic Division
- A shortcut method for dividing a polynomial by a linear divisor of the form x - a.
Common Mistakes to Avoid
- Using the wrong sign for the test value. If the divisor is x - 4, substitute x = 4, but if the divisor is x + 4, substitute x = -4.
- Thinking P(a) gives the quotient. The value P(a) gives only the remainder when dividing by x - a, not the full division result.
- Forgetting to include missing polynomial terms in synthetic division. Missing powers must be represented with coefficient 0 so the place values stay correct.
- Calling a number a root when the remainder is not zero. A value a is a root only when P(a) = 0, which means x - a is an actual factor.
Practice Questions
- 1 For P(x) = 2x^3 - 5x^2 + 3x - 7, find the remainder when P(x) is divided by x - 2.
- 2 Determine whether x + 3 is a factor of P(x) = x^3 + 2x^2 - 5x - 6.
- 3 Explain why the statement P(4) = 0 tells you both an algebra fact about P(x) and a graph fact about y = P(x).