Polynomial identities
A polynomial identity is true for every value of the variable, so both sides must have the same coefficient for each power. Expand structured expressions first, then solve the coefficient equations for the parameters
\[P(x)=Q(x)\text{ for all }x\Rightarrow\text{equal coefficients for equal powers}\]
- Expand structured expressions Expand brackets and powers before comparing the two sides \(\displaystyle (x+2)^2=x^2+4x+4\)
- Collect like powers Combine coefficients only when the powers of the variable are equal \(\displaystyle 3x^2+2x-x^2+5=2x^2+2x+5\)
- Equate equal powers An identity has the same coefficient for each power on both sides \(\displaystyle ax^2+bx+c=4x^2-3x+7\Rightarrow a=4,\ b=-3,\ c=7\)
- Solve parameter equations Use the coefficient equations to find the unknown parameters exactly \(\displaystyle 2p+1=7\Rightarrow p=3\)
Worked example
Find p and q if the following equality is a polynomial identity
\[3x^2+7x+2=p(x+2)^2+q(x+2)\]
- \(\displaystyle (x+2)^2=x^2+4x+4\)Expand the squared factor before comparing coefficients
- \(\displaystyle p(x+2)^2+q(x+2)=px^2+(4p+q)x+(4p+2q)\)Collect the quadratic, linear and constant coefficients
- \(\displaystyle p=3,\qquad 4p+q=7\Rightarrow q=7-12=-5\)Equate the quadratic and linear coefficients, then solve for the parameters
- \(\displaystyle 4p+2q=4(3)+2(-5)=2\)The constant coefficient also agrees, so the identity is satisfied
Common mistakes
Using one input only
✕\(\displaystyle P(1)=Q(1)\Rightarrow P(x)=Q(x)\)✓\(\displaystyle P(x)=Q(x)\text{ for all }x\Rightarrow\text{match every equal-power coefficient}\)
One matching input gives one equation, but an identity must hold for every inputEquating unlike powers
✕\(\displaystyle 2x^2+3x\Rightarrow 2+3=5\)✓\(\displaystyle 2x^2+3x\Rightarrow\text{coefficient of }x^2=2\text{ and coefficient of }x=3\)
Coefficients are combined or compared only when their powers are equalIncomplete expansion
✕\(\displaystyle (x+1)^2=x^2+1\)✓\(\displaystyle (x+1)^2=x^2+2x+1\)
The linear term is part of the square and must be retainedDifficulty levels
introCompare coefficients directly in an already expanded identity with two parameters
standardExpand named polynomial expressions before solving two coefficient equations
challengeSolve identities written with shifted powers or factor products and three parameters
examChoose a valid coefficient system, solve four parameters and verify the identity at an input
Questions about this topic
What makes an equality a polynomial identity?+
It is true for every value of the variable in the stated domain, not just for one or two selected values
Why are coefficients of equal powers compared?+
Two polynomials are equal for every input only when the coefficient of each power is the same on both sides after expansion
What happens if a power is missing?+
Its coefficient is zero, so the zero coefficient must be used when the corresponding powers are compared
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