How to divide one polynomial by another step by step, find the quotient and remainder, handle missing terms and check your answer.
Division with a remainder
Remember long division with whole numbers? Dividing 47 by 6 gives quotient 7 and remainder 5, because
\[47 = 6 \cdot 7 + 5 \qquad \text{and } 5 < 6\]
Polynomials divide the same way. The only change is that "smaller" now means lower degree.
The same fact is often written as a fraction: \(\dfrac{P(x)}{D(x)} = Q(x) + \dfrac{R(x)}{D(x)}\). When the remainder is zero, \(D(x)\) is a factor of \(P(x)\) and we say \(P(x)\) is divisible by \(D(x)\).
Getting set up
Step 2 is the one people skip. Without the placeholders, terms end up in the wrong columns and the answer goes wrong.
The algorithm
The process is a loop of four moves: divide, multiply, subtract, bring down. Let's divide \(3x^3 - 5x^2 + 4x - 7\) by \(x - 2\). The dividend is already in standard form and complete.
On paper this is laid out just like numerical long division: the divisor to the left of the division bracket, the quotient built up on top, and each product written under the dividend, lined up by degree.
Checking the answer
The definition gives you a built-in check: \(D \cdot Q + R\) must equal \(P\).
A quick sanity check: the quotient's degree is always the dividend's degree minus the divisor's degree. Here \(4 - 2 = 2\), and the quotient is indeed quadratic.
Practice
Quick check
A degree-6 polynomial is divided by a degree-2 polynomial. What's the degree of the quotient?
Subtract the degrees: \(6 - 2 = 4\).
Quick check
When do you stop dividing?
You stop once the leftover polynomial has a lower degree than the divisor. It may be zero, but it doesn't have to be.
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