Math · College algebra · Worked example
Divide polynomials with synthetic division
Divide 2x³ − 3x² − 11x + 6 by x − 3 using synthetic division, then find all the zeros of the polynomial.
Set up
The divisor x − 3 gives c = 3. The coefficients are 2, −3, −11 and 6.
Bring down, multiply, add
Bring down 2. Multiply by 3 and add: −3 + 6 = 3. Repeat: −11 + 9 = −2. Repeat: 6 + (−6) = 0.
| c = 3 | x³ | x² | x | 1 |
|---|---|---|---|---|
| Coefficients | 2 | −3 | −11 | 6 |
| Multiply by 3 | 6 | 9 | −6 | |
| Sums | 2 | 3 | −2 | 0 |
Read the result
The sums 2, 3 and −2 are the quotient’s coefficients, one degree lower, and the last sum, 0, is the remainder. So x − 3 is a factor.
Factor completely
Factor the quadratic quotient.
List the zeros
Each factor gives a zero: x = 3, x = 1/2 and x = −2.
Result
2x³ − 3x² − 11x + 6 = (x − 3)(2x² + 3x − 2) = (x − 3)(2x − 1)(x + 2), with zeros 3, 1/2 and −2.
Your turn
Divide x³ − 7x + 6 by x + 3 using synthetic division.
Show the answer and explanation
Quotient x² − 3x + 2, remainder 0.
Use c = −3 and the coefficients 1, 0, −7 and 6, with 0 for the missing x² term. The sums are 1, −3, 2 and 0.
Keep exploring
Graph plots the cubic with its three zeros marked.
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