1Cademy - Finding the Quotient $$(125x^3 - 8) \div (5x

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Inserting Placeholder Terms for Missing Degrees in Polynomial Division

Example

Finding the Quotient $(125x^3 - 8) \div (5x - 2)$

Apply polynomial long division to divide a two-term polynomial by a binomial whose leading coefficient is not $1$ , utilizing placeholders for multiple missing degree terms: $(125x^3 - 8) \div (5x - 2)$ .

Step 1 — Insert placeholders. The dividend $125x^3 - 8$ jumps from degree 3 directly to degree 0, meaning it is missing both the $x^2$ and $x$ terms. Rewrite it with placeholders: $125x^3 + 0x^2 + 0x - 8$ . Set this up under the division bracket with $5x - 2$ outside.

Step 2 — Divide $125x^3$ by $5x$ . The result is $25x^2$ . Write $25x^2$ in the quotient. Multiply $25x^2(5x - 2) = 125x^3 - 50x^2$ . Subtract $(125x^3 + 0x^2) - (125x^3 - 50x^2) = 50x^2$ . Bring down $0x$ to form $50x^2 + 0x$ .

Step 3 — Divide $50x^2$ by $5x$ . The result is $10x$ . Write $10x$ in the quotient. Multiply $10x(5x - 2) = 50x^2 - 20x$ . Subtract $(50x^2 + 0x) - (50x^2 - 20x) = 20x$ . Bring down $-8$ to form $20x - 8$ .

Step 4 — Divide $20x$ by $5x$ . The result is $4$ . Write $4$ in the quotient. Multiply $4(5x - 2) = 20x - 8$ . Subtract $(20x - 8) - (20x - 8) = 0$ .

The quotient is $25x^2 + 10x + 4$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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