Equations and inequalities

Polynomial division

Polynomial division works like long division with numbers. München names it for the T-Kurs entrance test, and you need it to finish a cubic equation once you have guessed one root.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the whole divisor by that result and subtract it from the dividend.
• Bring down the next term and repeat until the degree is smaller than the divisor's.
• Write missing powers with a zero coefficient: x³ − 1 = x³ + 0x² + 0x − 1.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Divide x³ − 4x² + x − 30 by (x − 5). What is the coefficient of x in the quotient?

   • a1
   • b6
   • c0
   • d2
   Show solution

   Answer: 1

   Since the division is exact, dividend = divisor × quotient. Divide term by term (or use synthetic division by r).

   x³ − 4x² + x − 30 = (x − 5)·(x² + x + 6), so the quotient is x² + x + 6.

2. 2. Divide x³ + 8x² + 15x + 5 by (x + 2). What is the remainder?

   • a-2
   • b0
   • c1
   • d-1
   Show solution

   Answer: -1

   Remainder theorem: dividing a polynomial by (x − r) leaves remainder P(r).

   By construction, x³ + 8x² + 15x + 5 = (x + 2)·(x² + 6x + 3) − 1, so the remainder is -1.

3. 3. Divide x³ − 19x + 12 by (x − 4). What is the coefficient of x in the quotient?

   • a-3
   • b4
   • c5
   • d3
   Show solution

   Answer: 4

   Since the division is exact, dividend = divisor × quotient. Divide term by term (or use synthetic division by r).

   x³ − 19x + 12 = (x − 4)·(x² + 4x − 3), so the quotient is x² + 4x − 3.

4. 4. Divide x³ + 2x² + 3x − 3 by (x − 1). What is the remainder?

   • a3
   • b4
   • c2
   • d-3
   Show solution

   Answer: 3

   Remainder theorem: dividing a polynomial by (x − r) leaves remainder P(r).

   By construction, x³ + 2x² + 3x − 3 = (x − 1)·(x² + 3x + 6) + 3, so the remainder is 3.

5. 5. Divide x³ + 2x² − 18x − 22 by (x − 4). What is the remainder?

   • a-2
   • b1
   • c2
   • d3
   Show solution

   Answer: 2

   Remainder theorem: dividing a polynomial by (x − r) leaves remainder P(r).

   By construction, x³ + 2x² − 18x − 22 = (x − 4)·(x² + 6x + 6) + 2, so the remainder is 2.

6. 6. Divide x³ − 4x² + 6x − 3 by (x − 1). What is the coefficient of x in the quotient?

   • a-2
   • b-4
   • c-3
   • d3
   Show solution

   Answer: -3

   Since the division is exact, dividend = divisor × quotient. Divide term by term (or use synthetic division by r).

   x³ − 4x² + 6x − 3 = (x − 1)·(x² − 3x + 3), so the quotient is x² − 3x + 3.

Common mistakes

• Forgetting a placeholder for a missing power.
• Subtracting only the first term instead of the whole product.
• Sign errors when subtracting a negative term.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs