The Polynomial Remainder Theorem
The Remainder Theorem says dividing a polynomial P(x) by x - a leaves a remainder equal to P(a). Plug a in, and you get the remainder without doing any division.
Divide any polynomial P(x) by a linear piece x - a, and you get a quotient Q(x) plus a leftover number r, so P(x) = (x - a)Q(x) + r. That r is just a number because the divisor has degree 1, so the remainder's degree has to be less than 1 - degree zero, a constant. Now plug in x = a. The term (a - a)Q(a) vanishes completely, no matter what Q looks like, and you're left with P(a) = r. That's the whole theorem: the remainder of dividing by x - a is always P(a). You never have to carry out the division to know what's left over.
This turns long division into a one-step evaluation. Long division by x - a grinds through subtraction after subtraction, each step peeling off one term, until a constant refuses to simplify further - that constant is the remainder, and the theorem tells you in advance it has to equal P(a). So instead of dividing x3 - 2x2 + 5x - 4 by x - 2, just compute P(2) = 8 - 8 + 10 - 4 = 6. The remainder is 6, found in four multiplications and no division at all.
The theorem's sharpest use is spotting factors. A remainder of 0 means (x - a)Q(x) + 0 = P(x), so x - a divides P(x) with nothing left over - that's the definition of a factor. So P(a) = 0 and 'x - a is a factor' are the same fact stated two ways, which is why this sits next to factoring and root-finding work on the test: testing a candidate root is just evaluating P at that number.
Key terms
- Remainder Theorem
- Dividing P(x) by x - a leaves a remainder equal to P(a).
- Factor Theorem
- x - a is a factor of P(x) exactly when P(a) = 0.
- Quotient
- The polynomial result of division, left over after removing the remainder.
- Degree
- The highest power of x in a polynomial, which limits how small a remainder's degree must be.
Watch it explained
- How it worksHow does long division by x - a always leave behind a constant that P(a) already knew?
- Worked exampleFind the remainder when x3 - 2x2 + 5x - 4 is divided by x - 2, using one evaluation.
- Common mistakeWhy isn't a remainder of zero a lucky coincidence rather than proof of a factor?
- Worked exampleFind whether x + 3 divides x3 + 2x2 - 5x + 6 evenly, using a single substitution.
Check yourself
End of topic.