The remainder in polynomial long division is what's left over after you've divided as much as possible. It's either zero, meaning the divisor goes in evenly, or a polynomial of lower degree than the divisor.
Here's a breakdown of what happens with the remainder:
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The Remainder Theorem & Factor Theorem: The remainder when a polynomial f(x) is divided by (x - a) is f(a). If the remainder is zero, then (x - a) is a factor of f(x). This is the factor theorem.
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Expressing the Result: You can express the result of the division as:
Dividend / Divisor = Quotient + Remainder / Divisor
Or, equivalently:
Dividend = (Divisor Quotient) + Remainder*
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Zero Remainder: If the remainder is zero, the divisor is a factor of the dividend. This means the division is "exact".
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Non-Zero Remainder: If the remainder is not zero, the divisor is not a factor of the dividend. The remainder is typically included as a fractional term added to the quotient, as shown in the equations above.
Example:
Let's say you divide x2 + 3x + 5 by x + 1.
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Perform the long division (which is not shown here for brevity).
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You find that the quotient is x + 2 and the remainder is 3.
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Therefore:
(x2 + 3x + 5) / (x + 1) = (x + 2) + 3/(x + 1)
Or, equivalently:
x2 + 3x + 5 = (x + 1)(x + 2) + 3
In summary, the remainder represents the portion of the dividend that the divisor couldn't "divide" evenly. You include it as the numerator of a fraction with the divisor as the denominator, and add this to the quotient.
