1Cademy - Finding the Quotient $$(x^3 - 64) \div (x

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Inserting Placeholder Terms for Missing Degrees in Polynomial Division

Example

Finding the Quotient $(x^3 - 64) \div (x - 4)$

Apply polynomial long division to divide a two-term polynomial by a binomial, utilizing placeholders for multiple missing degree terms: $(x^3 - 64) \div (x - 4)$ .

Step 1 — Insert placeholders. The dividend $x^3 - 64$ skips from degree 3 directly to degree 0, meaning it is missing both the $x^2$ and $x$ terms. Rewrite it in standard form with placeholders: $x^3 + 0x^2 + 0x - 64$ . Set this up under the division bracket with $x - 4$ outside.

Step 2 — Divide $x^3$ by $x$ . The result is $x^2$ . Write $x^2$ in the quotient. Multiply $x^2(x - 4) = x^3 - 4x^2$ and align it beneath the dividend. Subtract $(x^3 + 0x^2) - (x^3 - 4x^2) = 4x^2$ . Bring down $0x$ to get $4x^2 + 0x$ .

Step 3 — Divide $4x^2$ by $x$ . The result is $4x$ . Write $4x$ in the quotient. Multiply $4x(x - 4) = 4x^2 - 16x$ and write it below. Subtract $(4x^2 + 0x) - (4x^2 - 16x) = 16x$ . Bring down $-64$ to get $16x - 64$ .

Step 4 — Divide $16x$ by $x$ . The result is $16$ . Write $16$ in the quotient. Multiply $16(x - 4) = 16x - 64$ and write it below. Subtract $(16x - 64) - (16x - 64) = 0$ .

The quotient is $x^2 + 4x + 16$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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