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Apply the remainder and factor theorems

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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. 1Find 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. 2Evaluate 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. 3Use 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. 4Translate conditions for parameters Substitute the divisor zero into the polynomial when an unknown coefficient must be found \(\displaystyle P(x)=x^2+px+1,\quad P(2)=9\Rightarrow4+2p+1=9\Rightarrow p=2\)
Year 12, VCE Units 3 & 4, Chapter 1, 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 \(\displaystyle P(x)=x^2+px+1,\quad P(2)=9\Rightarrow4+2p+1=9\Rightarrow p=2\)

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
✕\(\displaystyle 2x+6=0\Rightarrow x=3\)✓\(\displaystyle 2x+6=0\Rightarrow x=-3\)
The divisor zero is found by solving the linear equation, so 2x=-6 gives x=-3
Using P(0) automatically
✕\(\displaystyle D(x)=x-4\Rightarrow R=P(0)\)✓\(\displaystyle D(x)=x-4\Rightarrow R=P(4)\)
The remainder theorem uses the zero of the divisor, and x-4 has zero 4
Missing the factor link
✕\(\displaystyle P(2)=0\not\Rightarrow x-2\text{ is a factor}\)✓\(\displaystyle P(2)=0\Longleftrightarrow x-2\text{ is a factor}\)
For a linear factor x-a, P(a)=0 and a zero remainder are equivalent conditions

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 theorem states that division by ux+v leaves the value P(a), where a is the solution of ux+v=0

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 is P(0) the remainder?+

For a divisor such as x or 3x, its zero is 0, so the theorem gives P(0). For any other linear divisor, evaluate at its own zero; the result may happen to equal P(0), but that equality 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 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations

In this chapter

Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems