Divide a Polynomial by a Linear Factor

Quick answer: Polynomial division must satisfy dividend = divisor × quotient + remainder. With a linear divisor, the remainder is constant. Include zero coefficients for missing powers when using synthetic division.

Worked example

Divide x³ − 2x² − 5x + 6 by x − 3.

1. Using root 3 and coefficients 1, −2, −5, 6: bring down 1.
2. Multiply by 3 and add to −2 to get 1.
3. Multiply that by 3 and add to −5 to get −2.
4. Multiply −2 by 3 and add to 6 to get 0. Quotient is x² + x − 2, remainder 0.

Check the result

(x − 3)(x² + x − 2) expands to x³ − 2x² − 5x + 6.

Common mistake to avoid

For x + 3, use synthetic value −3. Missing coefficients, such as the x term in x² + 1, must be entered as zero.

Try it yourself

1. Divide x² − 1 by x − 1. 2. Divide x² + 1 by x − 1.

Answers and reasoning

1. Quotient x + 1, remainder 0. 2. Quotient x + 1, remainder 2, because (x − 1)(x + 1) + 2 = x² + 1.

Need help with your own attempt? Ask a study question. AI answers require an account and paid credits. Check important steps with your textbook or teacher.