Math · College algebra · Worked example

Divide polynomials with long division

Divide x³ + 2x² − 5x + 7 by x² − x + 1.

Divide the leading terms

x³ ÷ x² = x. Multiply: x(x² − x + 1) = x³ − x² + x. Subtracting it from the dividend leaves 3x² − 6x + 7.

3x² ÷ x² = 3. Multiply: 3(x² − x + 1) = 3x² − 3x + 3. Subtracting leaves −3x + 4.

−3x + 4 has degree 1, lower than the divisor’s degree 2, so it is the remainder. The quotient is x + 3.

Divisor times quotient plus remainder gives the dividend.

\begin{array}{l} (x^{2} - x + 1) (x + 3) + (- 3 x + 4) \\ x^{3} + 2 x^{2} - 5 x + 7 \end{array}

Quotient x + 3 and remainder −3x + 4: x³ + 2x² − 5x + 7 = (x² − x + 1)(x + 3) − 3x + 4.

Divide 6x² + 5x − 1 by 2x + 3.

Show the answer and explanation

6x² ÷ 2x = 3x, and subtracting 3x(2x + 3) = 6x² + 9x leaves −4x − 1. Then −4x ÷ 2x = −2, and subtracting −2(2x + 3) = −4x − 6 leaves 5.

\begin{array}{l} (2 x + 3) (3 x - 2) + 5 \\ 6 x^{2} + 5 x - 1 \end{array}

Math checks the division statement: the expanded form matches the dividend.

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Make it concrete

Open the example inputs, change a value and keep a useful result on your board.

Your existing work stays on this device. Examples open as editable copies.