1Cademy - Finding the Quotient $$(x^4 - 7x^2 + 7x + 6) \div (x + 3)$$

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Inserting Placeholder Terms for Missing Degrees in Polynomial Division

Example

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

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

Step 1 — Insert placeholders. The dividend is missing an $x^3$ term. Rewrite it with a placeholder: $x^4 + 0x^3 - 7x^2 + 7x + 6$ . 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 $-7x^2$ to form $-3x^3 - 7x^2$ .

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 $(-3x^3 - 7x^2) - (-3x^3 - 9x^2) = 2x^2$ . 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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