IBHM 086 107.pdf


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4 Polynomials

4.2 Factor and remainder theorems
The remainder theorem
If a polynomial f(x) is divided by 1x ⫺ h2 the remainder is f(h).

Proof
We know that f1x2 ⫽ 1x ⫺ h2Q1x2 ⫹ R where Q(x) is the quotient and R is the
remainder.
For x ⫽ h, f1h2 ⫽ 1h ⫺ h2Q1h2 ⫹ R
⫽ 10 ⫻ Q1h2 2 ⫹ R
⫽R
Therefore, f1x2 ⫽ 1x ⫺ h2 Q1x2 ⫹ f1h2 .

The factor theorem
If f1h2 ⫽ 0 then 1x ⫺ h2 is a factor of f(x).
Conversely, if 1x ⫺ h2 is a factor of f(x) then f1h2 ⫽ 0.

Proof
For any function f1x2 ⫽ 1x ⫺ h 2Q1x2 ⫹ f1h2 .
If f1h2 ⫽ 0 then f1x2 ⫽ 1x ⫺ h2Q1x2.
Hence 1x ⫺ h2 is a factor of f(x).
Conversely, if 1x ⫺ h2 is a factor of f(x) then f1x2 ⫽ 1x ⫺ h2 Q1x2.
Hence f1h2 ⫽ 1h ⫺ h2Q1h2 ⫽ 0.

Example
Show that 1x ⫹ 52 is a factor of f1x2 ⫽ 2x3 ⫹ 7x2 ⫺ 9x ⫹ 30.
This can be done by substituting x ⫽ ⫺5 into the polynomial.
f1⫺52 ⫽ 21⫺52 3 ⫹ 71⫺52 2 ⫺ 91⫺52 ⫹ 30
⫽ ⫺250 ⫹ 175 ⫹ 45 ⫹ 30
⫽0
Since f1⫺52 ⫽ 0, 1x ⫹ 52 is a factor of f1x2 ⫽ 2x3 ⫹ 7x2 ⫺ 9x ⫹ 30.

This can also be done using synthetic division. This is how we would proceed if asked to
fully factorise a polynomial.

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