Key takeaway: The remainder theorem says that dividing a polynomial P(x) by x − c always leaves the remainder P(c), so one substitution replaces an entire long division.
Grade 11 – Algebra 2A-APR.B.2Expressions and Equations
Definition of Remainder Theorem
The remainder theorem ties polynomial division to plain evaluation. Divide any polynomial \(P(x)\) by a binomial of the form \(x-c\) and the leftover piece is a single number. That number is the output \(P(c)\).
Why care? Long division and synthetic division take several lines of work, while one substitution takes a few seconds. When a problem asks only for the leftover, the remainder theorem is the shortcut: put \(c\) in place of \(x\), simplify with care, and stop.
Formula and notation
Dividing \(P(x)\) by \(x-c\) produces a quotient \(Q(x)\) and a remainder \(R\), written as
\[P(x) = (x-c)\,Q(x) + R, \quad R = P(c).\]
Because the divisor has degree 1, \(R\) is a constant. A divisor like \(x+6\) is \(x-(-6)\), so its test number is \(c=-6\).
Key facts
- The leftover has degree 0 because the divisor has degree 1.
- If the remainder theorem gives 0, the binomial divides the polynomial exactly.
- To test x + 6 you evaluate at -6, the opposite of the constant.
- Synthetic division using the same c ends with this remainder in its last cell.
Remainder Theorem example
A teacher asks for the remainder when \(P(x)=x^4-3x^3+x-7\) is divided by \(x+2\). Use the remainder theorem instead of dividing.
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Rewrite \(x+2\) as \(x-(-2)\), which gives \(c=-2\).
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Evaluate: \(P(-2)=(-2)^4-3(-2)^3+(-2)-7\).
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That is \(16-3(-8)-2-7=16+24-2-7=31\).
The leftover after dividing is 31.
Common mistake
Watch out
Keiko wants the leftover of \(x^3+6x^2-4\) after dividing by \(x+2\). She substitutes 2 and writes \(8+24-4=28\).
Why it is wrong
The binomial \(x+2\) is \(x-(-2)\), so the number to plug in is \(-2\). The correct value is \(-8+24-4=12\). Flip the sign of the constant before substituting.
Related terms and lessons
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