Factor Theorem Calculator
Test if (x - c) is a factor and find complete polynomial factorization
Enter the polynomial to factor
Test if (x - 1) is a factor
Understanding the Factor Theorem
The Factor Theorem is a special case of the Remainder Theorem that helps you determine if a linear expression (x - c) is a factor of a polynomial. It's an essential tool for factoring polynomials and finding roots.
The Factor Theorem Statement
(x - c) is a factor of polynomial P(x) if and only if P(c) = 0
This means that if you substitute c into the polynomial and get 0, then (x - c) divides evenly into the polynomial with no remainder. Conversely, if P(c) ≠ 0, then (x - c) is not a factor.
How It Relates to the Remainder Theorem
The Remainder Theorem tells us that when dividing P(x) by (x - c), the remainder is P(c). The Factor Theorem is the special case where this remainder equals 0. When the remainder is 0, division is exact, making (x - c) a factor.
Steps to Use the Factor Theorem
- Identify c: Determine what value of c to test
- Evaluate P(c): Substitute c into the polynomial
- Check the result: If P(c) = 0, then (x - c) is a factor
- Find more factors: If it's a factor, use synthetic division to find quotient
- Repeat: Test more values to find all factors
Finding Potential Factors
Use the Rational Root Theorem to find potential values of c to test. The possible rational roots are ±(factors of constant term)/(factors of leading coefficient). Test these values systematically using the Factor Theorem.
Example
Test if (x - 1) is a factor of P(x) = x³ - 6x² + 11x - 6:
- We need to test c = 1
- P(1) = 1³ - 6(1²) + 11(1) - 6
- P(1) = 1 - 6 + 11 - 6 = 0
- Since P(1) = 0, (x - 1) IS a factor!
- We can now factor: P(x) = (x - 1)(x² - 5x + 6) = (x - 1)(x - 2)(x - 3)
Applications
- Factoring polynomials completely
- Finding roots and zeros of polynomials
- Solving polynomial equations
- Graphing polynomial functions (finding x-intercepts)
- Simplifying rational expressions
Frequently Asked Questions
What is the Factor Theorem?
The Factor Theorem states that (x - c) is a factor of polynomial P(x) if and only if P(c) = 0. This means you can test if a value is a root by simply evaluating the polynomial at that value. If you get 0, you've found a factor.
How is the Factor Theorem different from the Remainder Theorem?
The Remainder Theorem tells you the remainder when dividing by (x - c) is P(c). The Factor Theorem is the special case where P(c) = 0, meaning the remainder is 0 and (x - c) is a factor. They're closely related, with the Factor Theorem being a specific application of the Remainder Theorem.
How do I find values to test?
Use the Rational Root Theorem: possible rational roots are ±(factors of constant term) / (factors of leading coefficient). For x³ - 6x² + 11x - 6, test ±1, ±2, ±3, ±6. Not all will be roots, but any actual rational roots must be in this list.
What do I do after finding a factor?
Use synthetic division to divide the polynomial by the factor you found. This gives you a quotient of lower degree. Then continue testing values on the quotient to find more factors. Repeat until the polynomial is fully factored.
Can a polynomial have irrational or complex factors?
Yes! The Factor Theorem works for any value of c, including irrational numbers and complex numbers. However, the Rational Root Theorem only helps find rational roots. For irrational roots like √2, you might need other methods like the quadratic formula.
What if P(c) is very close to 0 but not exactly 0?
Due to rounding in calculations, you might get values very close to 0 (like 0.0000001) instead of exactly 0. If the value is extremely close to 0, it's likely a factor and the small difference is due to rounding. Always verify with synthetic division.
How many factors can a polynomial have?
A polynomial of degree n can have at most n linear factors (over complex numbers). For example, a cubic polynomial (degree 3) can have up to 3 linear factors. Some factors might be repeated, and some might be complex rather than real.
Can this help me solve polynomial equations?
Yes! If you need to solve P(x) = 0, the Factor Theorem helps you find the solutions (roots). Each factor (x - c) where P(c) = 0 gives you a solution x = c. Find all factors to find all solutions to the equation.
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