1Cademy - Finding the Quotient $$(2x^2 - 5x - 3) \div (x

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Dividing a Polynomial by a Binomial

Example

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

Apply polynomial long division to divide a trinomial with a non-unit leading coefficient by a binomial that involves subtraction: $(2x^2 - 5x - 3) \div (x - 3)$ .

Step 1 — Set up. Write the dividend $2x^2 - 5x - 3$ under the division bracket and the divisor $x - 3$ outside. The dividend is in standard form.

Step 2 — Divide $2x^2$ by $x$ . The result is $2x$ . Write $2x$ in the quotient above the $x$ term of the dividend.

Step 3 — Multiply and subtract. Multiply $2x(x - 3) = 2x^2 - 6x$ and align it under the dividend. Subtract $2x^2 - 6x$ from $2x^2 - 5x$ by changing signs and adding: $(2x^2 - 5x) - (2x^2 - 6x) = x$ . Bring down $-3$ to form $x - 3$ .

Step 4 — Divide $x$ by $x$ . The result is $1$ . Write $+1$ in the quotient above the constant term.

Step 5 — Multiply and subtract. Multiply $1(x - 3) = x - 3$ and write it below. Subtract $x - 3$ from $x - 3$ by changing signs and adding: the remainder is $0$ .

The quotient is $2x + 1$ .

Check: Multiply $(x - 3)(2x + 1) = 2x^2 + x - 6x - 3 = 2x^2 - 5x - 3$ ✓.

This example highlights two additional complexities compared to dividing a monic trinomial by a binomial with addition: the dividend's leading coefficient is $2$ (not $1$ ), and the divisor uses subtraction. When the divisor contains a minus sign, changing signs before adding at each subtraction step helps prevent errors.

Updated 2026-04-21

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OpenStax Elementary Algebra

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