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The Factor Theorem and Remainder Theorem are powerful tools for working with polynomials. They allow you to test for factors, find roots, and factorise cubics and higher-degree polynomials efficiently. Both are explicitly required by the Edexcel 9MA0 specification.
Statement: If a polynomial f(x) is divided by (x − a), the remainder is f(a).
More generally, if f(x) is divided by (ax − b), the remainder is f(b/a).
Example: Find the remainder when f(x) = 2x³ − 5x² + 3x + 7 is divided by (x − 2).
f(2) = 2(8) − 5(4) + 3(2) + 7 = 16 − 20 + 6 + 7 = 9.
The remainder is 9.
Example: Find the remainder when f(x) = 3x³ + x − 4 is divided by (3x − 1).
Evaluate f(1/3) = 3(1/27) + 1/3 − 4 = 1/9 + 1/3 − 4 = 1/9 + 3/9 − 36/9 = −32/9.
Statement: (x − a) is a factor of f(x) if and only if f(a) = 0.
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