Key takeaway: Synthetic division is a compact table method for dividing by x − r that uses only the coefficients and the number r to produce the quotient and a remainder.
Grade 11 – Algebra 2A-APR.D.6Expressions and Equations
Definition of Synthetic Division
Synthetic division boils polynomial division down to arithmetic on coefficients. It applies only when the divisor looks like \(x – 4\) or \(x + 2\): first power of \(x\), with a coefficient of exactly 1.
You write the zero of the divisor on the left, the coefficients of the dividend in a row, bring down the first coefficient, then repeatedly multiply by the zero and add. The last number is the remainder, and the others are the coefficients of the quotient, one degree lower than the dividend. The remainder equals the value of the polynomial at the zero.
Formula and notation
For synthetic division by \(x – r\), put \(r\) in the box; for divisor \(x + 2\), that means \(r = -2\):
\[ P(x) = (x – r)\,Q(x) + R, \qquad R = P(r) \]
Fill any missing power of \(x\) with a coefficient of 0 before starting.
Key facts
- The number in the box is the zero of the divisor, so x + 2 uses -2.
- Bring down, multiply, add, and repeat across the row.
- The last number is the remainder and equals P(r).
- A remainder of 0 shows that x – r is a factor of the polynomial.
Synthetic Division example
Divide \(3x^3 + 4x^2 – 17x – 6\) by \(x + 2\) using synthetic division.
-
Divisor \(x + 2\) means \(r = -2\). Coefficients: 3, 4, \(-17\), \(-6\). Bring down 3.
-
\(3 \times (-2) = -6\); \(4 + (-6) = -2\). Then \(-2 \times (-2) = 4\); \(-17 + 4 = -13\).
-
\(-13 \times (-2) = 26\); \(-6 + 26 = 20\). The last number is the remainder.
The quotient is \(3x^2 – 2x – 13\) with remainder 20, so the result is \(3x^2 – 2x – 13 + \frac{20}{x+2}\).
Common mistake
Watch out
Cora divides by \(x + 2\) using \(r = 2\) in the box and obtains a remainder of 4.
Why it is wrong
The box holds the zero of the divisor, the value that makes it 0. For \(x + 2\) that is \(-2\), not 2. With \(r = 2\) she is really dividing by \(x – 2\). Check with the remainder theorem: \(P(-2) = 3(-8) + 4(4) – 17(-2) – 6 = 20\).
Related terms and lessons
- Polynomial Long Division
- Remainder Theorem
- Factor Theorem
- Polynomial
- Zeros of a Function
- Rational Root Theorem
Browse the glossary: A–Z list · Algebra 2 vocabulary · Expressions and Equations