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Long Division of Polynomials

Polynomial long division is similar to long division of basic math, however polynomials have multiple terms.

How to Solve a Polynomial Division Math Problem

When we use long division math:

Quotient and Remainder

Divisor / Dividend

2 R 1

3 / 7

− 6

The answer is 2 + Remainder.

Remainder = 1/3, therefore the answer is 2 + 1/3 or 2 1/3.

To prove this:

3 (2 + 1/3) = 6 + 1 = 7

Use the same approach for division of polynomials.

To solve a polynomial division problem:

(2x3 − 7x + 2) / (x2 + x − 1)

f(x) = 2x3 − 7x + 2 (dividend)

g(x) = x2 + x − 1 (divisor)

q(x) = quotient and remainder

There are multiple terms to consider, proceed term-by-term.

The math term of the dividend having the highest degree is 2x3. Find a multiple of x2 equal or as close as possible to 2x3.

(x2) (2x) = 2x3

Write the term as the first term of the quotient:

x2 + x − 1 / 2x3 − 7x + 2

Next, multiply each term of the divisor by the 2x and write it beneath the dividend:

x2 + x − 1 / 2x3 − 7x + 2

2x3 + 2x2 − 2x

The divisor does not have a second degree term. Insert a zero term:

x2 + x − 1 / 2x3 + 0x2 − 7x + 2

−(2x3 + 2x2 − 2x)

−2x2 − 5x + 2

Subtract from divisor.

Apply the same steps to find a quotient term of −2x2:

(x2) (−2) = (−2x2)

Write the term as the second term of the quotient and multiply each term of the divisor by the (−2) and write it beneath the dividend:

2x − 2

x2 + x − 1 / 2x3 + 0x2 − 7x + 2

−(2x3 + 2x2 − 2x)

−2x2 − 5x + 2

−(−2x2 − 2x + 2)

−3x

Remainder

Thus, q(x) = 2x − 2 with a Remainder of −3x