Eleventh Grade · Math · week 15

Dividing polynomials

Divide polynomials using long division and synthetic division.

Dividing a polynomial by a linear factor works a lot like long division of numbers -- divide, multiply, subtract, bring down, repeat. Synthetic division is a streamlined shortcut for the special (and very common) case of dividing by (x - k): it uses only the coefficients, and it is faster once you trust it.

In synthetic division, write k outside and the polynomial's coefficients in a row (using 0 for any missing power). Bring down the first coefficient, multiply by k and add into the next column, and repeat across the row. The last number is the remainder; the rest are the coefficients of a quotient one degree lower than you started.

📋 For the teacher
Teach it (10–15 min):
  • Set up the synthetic division "box" together the same way every time (k outside, coefficients in a row, a blank row below for the multiply-and-add products) so the layout becomes automatic.
  • Emphasize the trap in the last try item out loud: dividing by (x+3) means using k = -3 in synthetic division, not +3 -- the sign flips.
  • Have them cross-check one result by multiplying the quotient by the divisor (and adding the remainder) to confirm it reconstructs the original polynomial.
  • Connect a remainder of 0 to factoring explicitly: it means the divisor is an exact factor, an idea that becomes the Remainder and Factor Theorems next week.
Talk about it:
  • Why does a missing power (like no x^2 term) need a placeholder 0 in synthetic division?
  • What does k represent when dividing by (x - k), and how do you find it if the divisor is (x + 5)?
  • How can you check that a synthetic division was done correctly?

Mastery looks like: They correctly set up and execute synthetic division by a linear factor, correctly identifying k (including when the divisor has a plus sign), and interpret a zero remainder.

Common stumbles: Using the wrong sign for k when the divisor is (x + a) instead of (x - a); forgetting a placeholder 0 for a missing power.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

DivisionShare fairly and build equal groups.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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