Long division of whole numbers was a major topic at school before the days of calculators and long division in algebra is a very similar process. You may have covered this earlier. The two processes are placed side by side below for comparison. See if you can follow the working.
| To calculate 3724 ÷ 15 | To calculate x3 + 3x2 + 6x + 5 ÷ (x + 1) | ||
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This means:
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This means:
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| Terminology | |||
| dividend | the expression being divided. | 3724 | x3 + 3x2 + 6x + 5 |
| divisor | the expression being divided in. | 15 | x + 1 |
| quotient | the answer | 248 | x2 + 2x + 4 |
| remainder | the left overs | 4 | 1 |
The Remainder Theorem
In the second example above, the remainder is 1.
This remainder can be worked out much quicker without the long division using the Remainder Theorem.
The Remainder Theorem states:
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If a polynomial f(x) is divided by (x − a) then the remainder is f(a).
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For the example above: When f(x) = x3 + 3x2 + 6x + 5
is divided by (x + 1) the remainder is f(-1)
Remainder is f(-1) = (-1)3 + 3(-1)2 + 6(-1) + 5 = 1
A much quicker way to find the remainder!
