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Provide the numerator and denominator polynomial, and the calculator will determine their remainder by using the remainder theorem.
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The Remainder Theorem Calculator is a free online tool that helps you quickly find the remainder when a polynomial is divided by a linear factor. It also provides step-by-step calculations for factoring and polynomial division, making it easier to understand the process.
In algebra, the remainder theorem (also called little Bézout’s theorem) states that when a polynomial f(x) is divided by a linear factor of the form x - j, the remainder is equal to f(j). This theorem is a direct application of Euclidean division of polynomials and was discovered by É Bézout.
Follow these steps to calculate the remainder by hand:
Find the remainder of \(f(x) = x^4 + 12x^3 + 18x^2 - 9x + 22\) when divided by \(x - 4\).
Solution:
Step 1: Identify j from the divisor x - 4 → j = 4
Step 2: Substitute j = 4 into f(x):
\(f(4) = 4^4 + 12(4^3) + 18(4^2) - 9(4) + 22\)
\(f(4) = 256 + 768 + 288 - 36 + 22\)
\(f(4) = 1298\)
Therefore, the remainder is 1298.
From Wikipedia: Polynomial Remainder Theorem – includes little Bézout’s theorem and the factor theorem.
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