Dividing Polynomials Synthetic Calculator
Answer
x^2 + 3 + (2)/(x - 2)
Polynomial Long & Synthetic Division
Quotient
x^2 + 3
Remainder
2
Remainder (numeric, constant remainders only)
2
Division algorithm check
x^3 - 2x^2 + 3x - 4 = (x - 2)(x^2 + 3) + (2)
Divides exactly
no
Degree of the quotient
2
Degree of the remainder
0
Method used
synthetic division
Synthetic division root
2
Worked out from OpenStax, College Algebra 2e, Section 5.4: Dividing Polynomials (long division, synthetic division, the Division Algorithm) and 1 other source.
How this is worked out
7 steps
- Divide x^3 - 2x^2 + 3x - 4 by x - 2.
- The divisor is linear, so use synthetic division with the root 2 (from x - 2 = 0).
- Bring down the leading coefficient, multiply by 2, add to the next coefficient, and repeat.
- Bottom row: 1, 0, 3, 2 - every entry but the last is a quotient coefficient, and the last entry is the remainder.
- Quotient x^2 + 3, remainder 2.
- Division algorithm: x^3 - 2x^2 + 3x - 4 = (x - 2)(x^2 + 3) + (2).
- Remainder Theorem: dividing by a linear factor leaves a constant remainder, so evaluating at the root gives it directly - p(2) = 2.
Quotient coefficients (highest power first)table
1, 0, 3
Remainder coefficients (highest power first)table
2
Synthetic division tableautable
| column | dividendCoefficient | carried | total |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | -2 | 2 | 0 |
| 3 | 3 | 0 | 3 |
| 4 | -4 | 6 | 2 |
Results are general information, not financial, tax, legal, medical or engineering advice.