You are here:
  1. / iCalculator
  2. / Equations
  3. / Identities
  4. / Eight Algebraic Identities

Menu

Math Lesson 9.3.4 - Eight Algebraic Identities

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

★★★★★ [ 1 Votes ]

Welcome to our Math lesson on Eight Algebraic Identities, this is the fourth lesson of our suite of math lessons covering the topic of Identities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Recalling the Eight Algebraic Identities

In tutorials 6.3, 8.1, 9.1 we discussed the eight algebraic identities that help us to simplify various algebraic expressions, including algebraic fractions. It is already known that the eight algebraic identities are:

  1. The square of a sum
    (a + b)2 = a2 + 2ab + b2
  2. The square of a difference
    (a - b)2 = a2 - 2ab + b2
  3. The difference of squares
    (a - b)(a + b) = a2 - b2
  4. The cube of a sum
    (a + b)3 = a3 + 3a2 b + 3ab2 + b3
  5. The cube of a difference
    (a - b)3 = a3 - 3a2 b + 3ab2 - b3
  6. The sum of cubes
    (a + b)(a2 - ab + b2 ) = a3 + b3
  7. The difference of cubes
    (a - b)(a2 + ab + b2 ) = a3 - b3
  8. The square of a sum of three terms
    (a + b + c)3 = a2 + b2 + c2 + 2ab + 2ac + 2bc

Actually, the identities in algebra are much more than eight but we have considered only the most representatives of them. For example, there is another algebraic identity that involves the square of the difference of three terms, i.e.

(a - b - c)3 = a2 + b2 + c2 - 2ab - 2ac + 2bc

Indeed,

(a - b - c)3
= [(a - b) - c]2
= (a - b)2 - 2(a - b) ∙ c + c2
= a2 - 2ab + b2 - 2ac + 2bc + c2
= a2 + b2 + c2 - 2ab - 2ac + 2bc

and so on.

The common feature of all the above algebraic identities (and many more) is that they are true for every value of the variables involved. Hence the name algebraic identities.

To make the distinction between an equation and an identity, we often use the symbol ( = ) for equations and ( ≡ ) for identities. Hence, the mathematical sentence

x3 - 8 ≡ (x - 2)(x2 + 4x + 4)

is an identity, as it contains the symbol ( ≡ ) to represent the relationship between the expressions in the two sides.

Example 2

Prove that (x - 2)(x2 + 2x + 4) = x3 - 8 is an identity.

Solution 2

All we need to do is to expand the expression on the left side and check whether all variables, coefficients and constants cancel out when sending everything on the same side. Thus, we have

(x - 2)(x2 + 2x + 4) = x3 - 8
x(x2 + 2x + 4) - 2(x2 + 2x + 4) = x3 - 8
x ∙ x2 + x ∙ 2x + x ∙ 4 - 2 ∙ x2 - 2 ∙ 2x - 2 ∙ 4 = x3 - 8
x3 + 2x2 + 4x - 2x2 - 4x - 8 = x3 - 8
x3 - 8 = x3 - 8

Now it is clear that we are dealing with an identity. However, let's complete our task properly, that is let's send everything on the left side. Thus,

x3 - 8 - x3 + 8 = 0
0x3 + 0 = 0
0x3 = 0

Now, we can confirm that

(x - 2)(x2 + 2x + 4) ≡ x3 - 8

[Beware of at the symbol ( ≡ ) we have used to express the identity.]

More Identities Lessons and Learning Resources

Equations Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
9.3Identities
Lesson IDMath Lesson TitleLessonVideo
Lesson
9.3.1Overview
9.3.2Identities, Conditional Identities (Equations) and Inconsistent Equations
9.3.3Identify Conditional Identities and Inconsistent Equations
9.3.4Eight Algebraic Identities
9.3.5Solving by Proof
9.3.6Rejecting a Supposition by Counter Example

Whats next?

Enjoy the "Eight Algebraic Identities" math lesson? People who liked the "Identities lesson found the following resources useful:

  1. Eight Algebraic Identities Feedback. Helps other - Leave a rating for this eight algebraic identities (see below)
  2. Equations Math tutorial: Identities. Read the Identities math tutorial and build your math knowledge of Equations
  3. Equations Video tutorial: Identities. Watch or listen to the Identities video tutorial, a useful way to help you revise when travelling to and from school/college
  4. Equations Revision Notes: Identities. Print the notes so you can revise the key points covered in the math tutorial for Identities
  5. Equations Practice Questions: Identities. Test and improve your knowledge of Identities with example questins and answers
  6. Check your calculations for Equations questions with our excellent Equations calculators which contain full equations and calculations clearly displayed line by line. See the Equations Calculators by iCalculator™ below.
  7. Continuing learning equations - read our next math tutorial: Iterative Methods for Solving Equations

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

★★★★★ [ 1 Votes ]

We hope you found this Math tutorial "Identities" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Equations Calculators by iCalculator™

© iCalculator™ 2009 - 2024 | Math Calculators | Privacy Policy | Terms of Use | Contact