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Apply the remainder and factor theorems

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Step 1 of 5

For the divisor below, which input should be used in the remainder theorem?

\[D(x)=2x+6\]
Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Remainder and factor theorems

For a linear divisor, first find the input that makes the divisor zero, then evaluate the polynomial at that input. A zero value is also the test for the divisor being a factor

\[D(x)=ux+v,\quad D(a)=0\Rightarrow a=-\frac{v}{u},\quad R=P(a)\]
  1. Find the divisor zero Set the linear divisor equal to zero and solve for its input value \(\displaystyle D(x)=2x+6=0\Rightarrow x=-3\)
  2. Evaluate at that input The remainder is the polynomial value at the zero of the divisor \(\displaystyle P(x)=x^2+2x+5,\quad R=P(-3)=9-6+5=8\)
  3. Use the factor condition A zero remainder means that the linear divisor is a factor of the polynomial \(\displaystyle R=0\Longleftrightarrow D(x)\text{ is a factor of }P(x)\)
  4. Translate conditions for parameters Substitute the divisor zero into the polynomial when an unknown coefficient must be found \(\displaystyle D(x)=x-2,\quad P(x)=x^2+px+1,\quad P(2)=9\Rightarrow4+2p+1=9\Rightarrow p=2\)

Worked example

Find the remainder when the polynomial is divided by the linear divisor, then decide whether the divisor is a factor

\[P(x)=x^3-2x^2-5x+6,\qquad D(x)=x-1\]
  1. \(\displaystyle x-1=0\Rightarrow x=1\)Find the zero of the divisor before evaluating the polynomial
  2. \(\displaystyle P(1)=1^3-2(1)^2-5(1)+6=1-2-5+6=0\)Substitute the divisor zero into every term of the polynomial
  3. \(\displaystyle R=P(1)=0\)The value at the divisor zero is the remainder
  4. \(\displaystyle R=0\Rightarrow x-1\text{ is a factor of }P(x)\)A zero remainder is exactly the factor condition for a linear divisor

Common mistakes

Reversing the divisor zero
✕\(\displaystyle 2x+6=0\Rightarrow x=3\)✓\(\displaystyle 2x+6=0\Rightarrow x=-3\)
Solve the linear equation: subtract six, then divide by two to get negative three
Using P(0) automatically
✕\(\displaystyle D(x)=x-4\Rightarrow R=P(0)\)✓\(\displaystyle D(x)=x-4\Rightarrow R=P(4)\)
Use the zero of the divisor, which is 4
Missing the factor link
✕\(\displaystyle P(2)=0\not\Rightarrow x-2\text{ is a factor}\)✓\(\displaystyle P(2)=0\Longleftrightarrow x-2\text{ is a factor}\)
A polynomial value of zero at the divisor zero is equivalent to a zero remainder

Difficulty levels

introFind a remainder with a monic linear divisor and an integer divisor zero
standardUse a non-monic divisor to give a factor or non-factor verdict
challengeRecover one or two coefficient parameters from factor and remainder conditions
examChoose the theorem procedure, state the factor result and verify at another input

Questions about this topic

Why is the polynomial evaluated at the divisor zero?+

The remainder is the polynomial's value at the zero of the linear divisor

How does a zero remainder show that a divisor is a factor?+

A zero remainder means the division is exact, so the divisor appears as a factor of the polynomial

When does evaluating at zero give the remainder?+

When the divisor has zero 0, as with a divisor of x or 3x. For another linear divisor, evaluate at its own zero; its remainder might happen to equal the value at 0, but that is not the rule

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