Teach Me Maths

Identities and Comparing Coefficients

How to distinguish identities from equations and find unknown constants by expansion, coefficient comparison and useful substitutions.

An identity holds for every allowed input

The symbol \equiv means that two expressions are identical: they have the same value for every allowed input. An equation such as x+3=8 is true only for particular values of x

For polynomial identities, expand and collect both sides. Coefficients of matching powers must agree, including the constant term. A missing power has coefficient zero.

Worked example 1

Find A and B if A(x-3)+B\equiv5x-11

  1. Expand the left side: Ax-3A+B. Comparing the x coefficients gives A=5

  2. Your turn. Use -3(5)+B=-11. Find B

    -15+B=-11, so B=4

  3. Answer: A=5, B=4. Expanding 5(x-3)+4 gives 5x-11

Explore matching coefficients

Try it: set A=1 and B=-1. Matching the curves at one input is not enough: at x=0, both always give -6, whatever A and B are.

Worked example 2

Find a and b if (x-2)(ax+b)\equiv3x^2-11x+10

  1. Expand: ax^2+(b-2a)x-2b. The x^2 coefficients give a=3

  2. Your turn. Compare the x coefficients: b-6=-11. Find b

    b=-5

  3. Check the constant: -2(-5)=10, as required.
  4. Answer: a=3, b=-5

Worked example 3

Find a and b if (x+2)(x^2+ax+b)\equiv x^3+5x^2+2x-8

  1. Expand and group by powers: x^3+(a+2)x^2+(b+2a)x+2b
  2. The x^2 coefficients give a+2=5, so a=3

  3. Your turn. Compare the constants: 2b=-8. Find b

    b=-4

  4. Answer: a=3, b=-4. Check the x coefficient: -4+2(3)=2

Worked example 4

Find p and q if 2x^2-12x+23\equiv2(x-p)^2+q

  1. Expand the right side: 2x^2-4px+2p^2+q

  2. Your turn. Compare the x coefficients: -4p=-12. Find p

    p=3

  3. The constants give 2(3)^2+q=23, so q=5
  4. Answer: p=3, q=5

Choose substitutions that remove a term

An identity is true at every allowed input, so a convenient substitution can make a bracket zero. Use enough information to determine every constant, then check by expansion. One numerical match alone does not establish an identity.

Worked example 5

Find A and B if 3x^2+6x-17\equiv A(x-1)(x+3)+B

  1. Set x=1. The bracket x-1 is zero, leaving B=3+6-17

  2. Your turn. Work out B

    B=-8

  3. Set x=0: -17=-3A-8, so A=3
  4. Answer: A=3, B=-8. Expanding gives 3(x^2+2x-3)-8=3x^2+6x-17

Worked example 6

Find A, B and C if (x+2)(x-3)(x+1)\equiv x^3+Ax^2+Bx+C

  1. The first two brackets give x^2-x-6. Multiply by x+1: x^3+x^2-x^2-x-6x-6

  2. Your turn. What is the coefficient of x^2 after collecting?

    The x^2 terms cancel, so A=0

  3. Answer: A=0, B=-7, C=-6. A missing term still has a coefficient to compare.

Common mistakes

  • Solving an identity for a special input. Find constants that make it true for every allowed input.
  • Comparing coefficients too early. Expand and collect like terms on both sides first.
  • Missing constants or zero coefficients. Compare the constant terms and include missing powers with coefficient zero.
  • Using one numerical match as proof. Show algebraically that the expressions agree for every allowed value.
  • Checking only some coefficients. After finding the constants, check every coefficient agrees.

Now try it: Identities and Comparing Coefficients practice questions

More on this topic: Identities and Comparing Coefficients worksheet with full solutions