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Divide polynomials and state quotient and remainder

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Step 1 of 5

A power is missing from the dividend. What should happen before long division?

Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Polynomial division

Align descending powers, divide leading terms, subtract the complete divisor multiple and repeat. The divisor must be a nonzero polynomial. Stop when the remainder is zero or has lower degree than the divisor, and check the division identity

\[D(x)=V(x)Q(x)+R(x),\qquad V(x)\not\equiv0,\qquad R(x)=0\text{ or }\deg R<\deg V\]
  1. Keep zero columns Write a zero coefficient for every missing power before aligning the division \(\displaystyle D(x)=x^3+0x^2-4x+8\)
  2. Divide leading terms Divide the leading term of the current dividend by the leading term of the divisor \(\displaystyle \frac{x^3}{x}=x^2\)
  3. Subtract the full multiple Multiply the whole divisor by the new quotient term, then subtract every term \(\displaystyle x^3+0x^2-4x+8-x^2\left(x-2\right)=2x^2-4x+8\)
  4. Check the identity The divisor must be a nonzero polynomial. Stop when the remainder is zero or has lower degree than the divisor, and check the division identity \(\displaystyle D(x)=V(x)Q(x)+R(x)\)

Worked example

Divide the displayed dividend by the displayed divisor and state the quotient and remainder

\[\frac{2x^3+0x^2-5x+6}{x-1}\]
  1. \(\displaystyle D(x)=2x^3+0x^2-5x+6,\quad V(x)=x-1\)Write the zero coefficient for the missing squared term so all powers remain aligned
  2. \(\displaystyle \frac{2x^3}{x}=2x^2,\quad D(x)-2x^2V(x)=2x^2-5x+6\)The first quotient term comes from dividing the leading terms, and subtracting its full divisor multiple leaves the next dividend
  3. \(\displaystyle \frac{2x^2}{x}=2x,\quad (2x^2-5x+6)-2xV(x)=-3x+6\)The next quotient term comes from dividing the new leading terms, and the subtraction leaves the displayed linear expression
  4. \(\displaystyle \frac{-3x}{x}=-3,\quad (-3x+6)-(-3)V(x)=3,\quad Q(x)=2x^2+2x-3,\quad R(x)=3\)The remainder is 3, whose degree is lower than the linear divisor

Common mistakes

Forgetting a zero column
✕\(\displaystyle x^3-4x+8\)✓\(\displaystyle x^3+0x^2-4x+8\)
A zero coefficient keeps every power in its correct division column
Adding instead of subtracting
✕\(\displaystyle 2x^3+0x^2-5x+6+2x^2\left(x-1\right)=4x^3-2x^2-5x+6\)✓\(\displaystyle 2x^3+0x^2-5x+6-2x^2\left(x-1\right)=2x^2-5x+6\)
Each divisor multiple must be subtracted from the current dividend
Leaving a high-degree remainder
✕\(\displaystyle D(x)=(x-1)(2x^2+2x)+(-3x+6)\)✓\(\displaystyle D(x)=(x-1)(2x^2+2x-3)+3\)
The remainder must have lower degree than the divisor, which is linear here

Difficulty levels

introIdentify quotient and remainder for a simple polynomial division
standardDivide a polynomial with a missing power and state the quotient and remainder
challengeDivide by a non-monic linear or quadratic divisor and check the identity
examChoose the valid next subtraction row for a linear divisor, then state every quotient and remainder coefficient, including exact-division cases

Questions about this topic

Why must a missing power be written with coefficient zero?+

The zero coefficient preserves the position of every other power. This keeps the subtraction columns aligned

When does polynomial long division stop?+

Stop when the current remainder is zero or has lower degree than the nonzero divisor. Zero remainder means the division is exact

How can a polynomial division answer be checked?+

Substitute the quotient and remainder into the displayed division identity and confirm that the identity expands to the original dividend

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials

In this chapter

Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model