2.4.3

Remainder Theorem

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Remainder Theorem

Dividing polynomials can be a tricky process. The remainder theorem allows us to quickly find the remainder of a polynomial when it is divided by a binomial.

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Division

  • A given polynomial can be decomposed into a quotient, divisor and remainder as follows:
    • f(x)=q(x)(xc)+rf(x) = q(x)(x-c) + r
  • Where f(x)f(x) is the polynomial, q(x)q(x) is the quotient, xcx-c is the divisor and rr is the remainder.
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Division

  • When evaluated for the value cc, this decomposition can be written as:
    • f(c)=q(c)(cc)+r=rf(c) = q(c)(c-c) + r = r.
  • This leads us to the remainder theorem.
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Remainder theorem

  • If the polynomial function f(x)f(x) is divided by xcx-c, then the remainder is f(c)f(c).
  • We can use this to find the remainder of polynomials for given devisors without actually having to do the division!
  • f(c)=0f(c) = 0 if and only if xcx-c is a factor of f(x)f(x).

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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