Remainder Theorem Calculator
Find polynomial remainder without division by evaluating P(c)
Enter the polynomial
For (x - 3), enter 3. For (x + 2), enter -2
Understanding the Remainder Theorem
The Remainder Theorem is a powerful shortcut that allows you to find the remainder when dividing a polynomial by (x - c) without actually performing division. Simply evaluate the polynomial at x = c, and that value is your remainder.
The Remainder Theorem Statement
If a polynomial P(x) is divided by (x - c), then the remainder is P(c).
In other words: Remainder = P(c)
Why It Works
When you divide P(x) by (x - c), you can write:
P(x) = (x - c) × Q(x) + R
where Q(x) is the quotient and R is the remainder (a constant). If you substitute x = c into this equation:
P(c) = (c - c) × Q(c) + R = 0 × Q(c) + R = R
Therefore, P(c) equals the remainder!
How to Use the Remainder Theorem
- Identify c: From divisor (x - c), determine the value of c
- Substitute: Replace every x in P(x) with the value c
- Evaluate: Calculate P(c) using order of operations
- Result: The value of P(c) is the remainder
Example
Find the remainder when P(x) = x³ - 4x² + 5x - 2 is divided by (x - 3):
- The divisor is (x - 3), so c = 3
- Evaluate P(3) = 3³ - 4(3²) + 5(3) - 2
- P(3) = 27 - 36 + 15 - 2 = 4
- Therefore, the remainder is 4
Connection to Factor Theorem
The Remainder Theorem leads directly to the Factor Theorem: if P(c) = 0 (remainder is 0), then (x - c) is a factor of P(x). This makes the Remainder Theorem invaluable for factoring polynomials and finding roots.
Frequently Asked Questions
What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder equals P(c). This means you can find the remainder by simply evaluating the polynomial at x = c, without performing any division.
Why use the Remainder Theorem instead of division?
The Remainder Theorem is much faster than polynomial division when you only need the remainder. Instead of doing long division or synthetic division, you just substitute one value and evaluate. It's particularly useful for testing whether (x - c) is a factor.
Does this work for divisors like (x + 5)?
Yes! Rewrite (x + 5) as (x - (-5)). So c = -5. The theorem works for any linear divisor of the form (x - c), where c can be positive, negative, a fraction, or a decimal. Just identify c correctly and evaluate P(c).
What if P(c) = 0?
If P(c) = 0, the remainder is 0, which means (x - c) divides evenly into P(x). This means (x - c) is a factor of P(x), and c is a root (or zero) of the polynomial. This is the Factor Theorem, which is a special case of the Remainder Theorem.
Can I use this to find the quotient?
No, the Remainder Theorem only gives you the remainder, not the quotient. If you need the quotient, you must use polynomial long division or synthetic division. However, the Remainder Theorem is perfect when you only care about the remainder.
Does this work for divisors of degree 2 or higher?
No, the Remainder Theorem only works for linear divisors (x - c). For divisors like x² + 1 or other higher-degree polynomials, you must use polynomial long division. The theorem specifically requires a divisor of the form (x - c).
How is this useful in real applications?
The Remainder Theorem is used to quickly test potential roots of polynomials, factor polynomials efficiently, solve polynomial equations, verify division results, and in coding theory for error detection. It's a fundamental tool in algebra and beyond.
Can I verify the result?
Yes! Perform the actual division using synthetic division or long division and check that you get the same remainder. Or use the division algorithm: verify that P(x) = (x - c) × Q(x) + R, where R is your calculated remainder.
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