ACT Math: Synthetic Division Divides Polynomials in Half the Time

Long polynomial division (traditional) works for all divisors but is slow and error-prone. Synthetic division works only for divisors of the form (x-a) or (x+a), but it is 3 times faster. On the ACT, nearly all polynomial division uses linear divisors, so synthetic division is almost always the better choice. If a question asks you to divide a polynomial by (x-3) or (x+2), use synthetic division to save 2 minutes. The process uses only the coefficients and performs pure arithmetic—no algebra steps.

Example: Divide x³+2x²-5x+6 by (x-2). Traditional long division takes 5-7 minutes. Synthetic division takes 90 seconds.

Step 1: Write the coefficients of the dividend in a row. For x³+2x²-5x+6, write: [1, 2, -5, 6]. Step 2: Write the value 'a' from the divisor (x-a) to the left. For (x-2), write 2. Step 3: Bring down the first coefficient. Step 4: Multiply the brought-down number by 'a', write the result under the next coefficient, add, and repeat. For x³+2x²-5x+6 divided by (x-2): Bring down 1. 1×2=2; 2+2=4. 4×2=8; -5+8=3. 3×2=6; 6+6=12. Result: Quotient x²+4x+3, remainder 12. The last number is the remainder; all others are quotient coefficients.

Practice the layout until it feels natural. The key is vertical alignment and not skipping the multiply-add cycle.

Problem 1: Divide 2x³+3x²-8x+5 by (x+1). Coefficients: [2, 3, -8, 5]; a=-1. Bring 2. 2×(-1)=-2; 3+(-2)=1. 1×(-1)=-1; -8+(-1)=-9. -9×(-1)=9; 5+9=14. Quotient: 2x²+x-9, remainder 14. Problem 2: Divide x⁴-1 by (x-1). Coefficients: [1, 0, 0, 0, -1]; a=1. Bring 1. 1×1=1; 0+1=1. 1×1=1; 0+1=1. 1×1=1; 0+1=1. 1×1=1; -1+1=0. Quotient: x³+x²+x+1, remainder 0 (divisible!). Problem 3: Divide 3x²+7x-20 by (x+4). Do these three daily until you complete each in under 60 seconds.

Verify your answer by expanding (divisor × quotient + remainder) and checking it equals the original polynomial.

Polynomial division questions appear in 1-2 ACT Math sections, usually medium difficulty. Synthetic division is the fastest approach, and mastering it frees up time for harder questions. A student who knows synthetic division solves these questions in 1 minute; a student who only knows long division takes 5-7 minutes. That saved time is buffer for guessing or reworking uncertain problems.

Spend one study session mastering the process. Review it one day before the test. On test day, synthetic division becomes your secret fast-math weapon.