LEMMA

Advanced algebra

Polynomial division

ALEKS placement

Long division works on polynomials, and synthetic division is the shortcut for linear divisors.

What this covers

  • Write both polynomials in descending order with zero placeholders for missing powers.
  • Divide the leading terms to get the next quotient term.
  • Multiply back, subtract, and bring down.
  • For synthetic division use the root of the divisor, not its constant.

Worked example

Worked example

Divide x³ - 4x + 1 by x - 2.

  1. 10−41Include a 0 for the missing x² term.
  2. 1201Synthetic division with 2: bring down, multiply, add.
  3. x2+2x+x−21​Quotient x² + 2x, remainder 1.
Another worked example

Divide (x² − 7x + 12) by (x − 3).

  1. x−3x2−7x+12​=x−3(x−3)(x−4)​Factor the top: −3 and −4 multiply to 12 and add to −7.
  2. x−4,x=3The divisor is a factor of the top, so it cancels with no remainder. The original still excludes x = 3.

A common mistake

A mistake Lemma catches

From

r=1r=−1

“Synthetic division by x + 1 uses the number that makes x + 1 zero: −1 + 1 = 0, so r = −1. Using +1 divides by x − 1 instead: 1 + 1 = 2, not 0.”

Try one

Sample problem

Divide and simplify.

x−6x2−13x+42​

Practice polynomial division free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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