1Cademy - Finding the Quotient $$(x^3 - x^2 + x + 4) \div (x + 1)$$

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Dividing a Polynomial by a Binomial

Example

Finding the Quotient $(x^3 - x^2 + x + 4) \div (x + 1)$

Use polynomial long division to divide a cubic polynomial by a binomial, producing a quotient with a nonzero remainder: $(x^3 - x^2 + x + 4) \div (x + 1)$ .

Step 1 — Set up the long division. Write the dividend $x^3 - x^2 + x + 4$ under the division bracket and the divisor $x + 1$ outside. Confirm the dividend is in standard form with no missing degree terms.

Step 2 — Divide $x^3$ by $x$ . The result is $x^2$ . Write $x^2$ in the quotient above the $x^2$ term of the dividend. Multiply $x^2(x + 1) = x^3 + x^2$ and write it beneath the first two terms of the dividend.

Step 3 — Subtract and bring down. Subtract $x^3 + x^2$ from $x^3 - x^2$ by changing signs and adding: $(x^3 - x^2) - (x^3 + x^2) = -2x^2$ . Bring down the next term to get $-2x^2 + x$ .

Step 4 — Divide $-2x^2$ by $x$ . The result is $-2x$ . Write $-2x$ in the quotient. Multiply $-2x(x + 1) = -2x^2 - 2x$ and write it below.

Step 5 — Subtract and bring down. Subtract $-2x^2 - 2x$ from $-2x^2 + x$ : the result is $3x$ . Bring down the last term to get $3x + 4$ .

Step 6 — Divide $3x$ by $x$ . The result is $3$ . Write $3$ in the quotient. Multiply $3(x + 1) = 3x + 3$ and write it below.

Step 7 — Subtract to find the remainder. Subtract $3x + 3$ from $3x + 4$ : the remainder is $1$ . Since this remainder is nonzero but has a smaller degree than the divisor, the division is complete.

Step 8 — Express the remainder as a fraction. Write the remainder over the divisor: $\frac{1}{x + 1}$ .

The quotient is $x^2 - 2x + 3 + \frac{1}{x + 1}$ .

To verify, multiply $(x + 1)\left(x^2 - 2x + 3 + \frac{1}{x + 1}\right)$ ; the result should equal $x^3 - x^2 + x + 4$ . This example demonstrates that when a polynomial does not divide evenly by a binomial, the leftover amount (the remainder) is written as a fraction with the divisor as the denominator — similar to how whole-number division can produce a remainder.

Updated 2026-04-21

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OpenStax Elementary Algebra

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