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Algebraic division
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You can divide a polynomial by a linear expression just like long division with numbers.
Long division
Divide x3 + 2x2 − 5x − 6 by (x + 1):
x3 ÷ x = x2. Multiply: x3 + x2. Subtract: x2 − 5x.
x2 ÷ x = x. Multiply: x2 + x. Subtract: −6x − 6.
−6x ÷ x = −6. Multiply: −6x − 6. Subtract: 0.
Quotient x2 + x − 6, remainder 0.
x3 ÷ x = x2. Multiply: x3 + x2. Subtract: x2 − 5x.
x2 ÷ x = x. Multiply: x2 + x. Subtract: −6x − 6.
−6x ÷ x = −6. Multiply: −6x − 6. Subtract: 0.
Quotient x2 + x − 6, remainder 0.
Quotient and remainder
f(x) = (divisor) × (quotient) + remainder
When you divide by a linear expression, the remainder is a number.
If the remainder is 0, the divisor is a factor.
When you divide by a linear expression, the remainder is a number.
If the remainder is 0, the divisor is a factor.
Dividing by (ax + b)
The method is the same. Divide 6x3 + x2 − 4x + 1 by (2x − 1):
6x3 ÷ 2x = 3x2, then 4x2 ÷ 2x = 2x, then −2x ÷ 2x = −1
Quotient 3x2 + 2x − 1, remainder 0.
6x3 ÷ 2x = 3x2, then 4x2 ÷ 2x = 2x, then −2x ÷ 2x = −1
Quotient 3x2 + 2x − 1, remainder 0.
Divide 2x3 − 3x2 − 11x + 6 by (x − 3).
- 2x3 ÷ x = 2x2: subtract 2x3 − 6x2 to leave 3x2 − 11x
- 3x2 ÷ x = 3x: subtract 3x2 − 9x to leave −2x + 6
- −2x ÷ x = −2: subtract −2x + 6 to leave 0
Answer: 2x2 + 3x − 2
Divide the leading terms, multiply back, subtract, repeat. f(x) = divisor × quotient + remainder.
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