1Cademy - Finding the Quotient $$(x^4 - 11x^2 - 7x

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Inserting Placeholder Terms for Missing Degrees in Polynomial Division

Example

Finding the Quotient $(x^4 - 11x^2 - 7x - 6) \div (x + 3)$

Apply polynomial long division to divide a degree-four polynomial by a binomial, utilizing a placeholder for a missing term: $(x^4 - 11x^2 - 7x - 6) \div (x + 3)$ .

Step 1 — Insert placeholders. The dividend is missing an $x^3$ term. Rewrite it with a placeholder: $x^4 + 0x^3 - 11x^2 - 7x - 6$ . Set this up under the division bracket with $x + 3$ outside.

Step 2 — Divide $x^4$ by $x$ . The result is $x^3$ . Write $x^3$ in the quotient. Multiply $x^3(x + 3) = x^4 + 3x^3$ . Subtract $(x^4 + 0x^3) - (x^4 + 3x^3) = -3x^3$ . Bring down $-11x^2$ to form $-3x^3 - 11x^2$ .

Step 3 — Divide $-3x^3$ by $x$ . The result is $-3x^2$ . Write $-3x^2$ in the quotient. Multiply $-3x^2(x + 3) = -3x^3 - 9x^2$ . Subtract $(-3x^3 - 11x^2) - (-3x^3 - 9x^2) = -2x^2$ . Bring down $-7x$ to form $-2x^2 - 7x$ .

Step 4 — Divide $-2x^2$ by $x$ . The result is $-2x$ . Write $-2x$ in the quotient. Multiply $-2x(x + 3) = -2x^2 - 6x$ . Subtract $(-2x^2 - 7x) - (-2x^2 - 6x) = -x$ . Bring down $-6$ to form $-x - 6$ .

Step 5 — Divide $-x$ by $x$ . The result is $-1$ . Write $-1$ in the quotient. Multiply $-1(x + 3) = -x - 3$ . Subtract $(-x - 6) - (-x - 3) = -3$ . This is the remainder.

Step 6 — Express the remainder as a fraction. Write the remainder over the divisor: $-\frac{3}{x + 3}$ .

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

Updated 2026-04-29

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OpenStax Intermediate Algebra

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