Algebraic Identities

The history of algebraic identities begins with ancient civilizations like the Babylonians and Egyptians, who solved equations using words and basic methods such as false position. The Babylonians were especially advanced, handling cubic and quadratic equations using addition and multiplication. Al-Khwarizmi, an Islamic mathematician, is known as the father of algebra and played a key role in developing algebraic equations.

The modern form of algebra, including the use of symbols and variables, was further developed during the Medieval and Renaissance periods by mathematicians like René Descartes and Pierre de Fermat, who introduced modern notation and alphabets as variables.

The LHS (left-hand side) and RHS (right-hand side) of algebraic identities are the same. The identities remain constant even after the value changes. Algebraic identities consist of variables, numbers, and operations. Some standard algebraic identities are:
 

• \((a + b)^2 = a^2 + 2ab + b^2\)
   
• \((a - b)^2 = a^2 - 2ab + b^2\)
   
• \(a^2 - b^2 = (a + b)(a - b)\)
   
• \((x + a)(x + b) = x^2 + (a + b)x + ab\)

Let us take a look at the algebraic identities formula of some of the common algebraic identities in this section.

Some of the important algebraic formula identities that we will be using are given as;

1. Square identities

\((a + b)^2 = a^2 + 2ab + b^2 \)

\((a - b)^2 = a^2 - 2ab + b^2 \)

\((a + b)(a - b) = a^2 - b^2 \)

2. Cube identities

\((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)

\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)

3. Sum and difference of cubes

\(a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

\(a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

4. Square of trinomial

\((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \)

5. Cube of trinomial

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) \)

\((a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + a^2c + b^2a + b^2c + c^2a + c^2b) + 6abc \)

6. Product of two binomials

\((x + a)(x + b) = x^2 + (a + b)x + ab \)

\((x - a)(x - b) = x^2 - (a + b)x + ab \)

7. Special expansion and factorization

\(a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2) \)

\(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \)

8. General identity for any number of terms

\((x_1 + x_2 + x_3 + \dots + x_n)^2 = \sum_{i=1}^{n} x_i^2 + 2\sum_{i<j} x_i x_j \)

Algebraic identities make it easier for students to solve algebraic equations. In this section, let’s learn the types of all algebraic identities. 

Square of a binomial: The binomial expression has only two terms in it. The square of a binomial is squaring a sum or difference of two terms.

That is, \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\)

Difference of squares: The difference of squares is the difference between the perfect squares.

That is, \(a^2 - b^2 = (a + b) (a - b)\)

Cube of a binomial: The cube of a binomial is the addition or subtraction of the third power of the binomial terms.

The formulas are:

\((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)

Sum of cubes: The sum of cubes is the sum of two perfect cubes.

They are in the form, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Difference of cubes: The difference between two perfect cubes; which means \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Perfect square trinomial: The result of squaring a binomial is of three terms, which is a perfect square trinomial.

That is, \(a^2 + 2ab + b^2 = (a + b)^2\)

Proof of algebraic identities helps us in knowing why the identities are true and how it came to be like that. Let use try to prove the algebraic identities.

1. Proof of \(\pmb{(a + b)^2 = a^2 + 2ab + b^2}\)

\((a + b)^2 = (a + b)(a + b) \\[1em] \text{Expand the equation,}\\[1em] (a + b)^2= a(a + b) + b(a + b) \\[1em] (a + b)^2= a^2 + ab + ba + b^2 \\[1em] \text{Since} \ ab=ba,\\[1em] (a + b)^2= a^2 + 2ab + b^2\)

Hence, proved.

2. Proof of \(\pmb{(a - b)^2 = a^2 - 2ab + b^2}\)

\((a - b)^2 = (a - b)(a - b) \\[1em] \text{Expand the equation,}\\[1em] (a - b)^2= a(a - b) - b(a - b) \\[1em] (a - b)^2= a^2 - ab - ba + b^2 \\[1em] \text{Since}\ ab=ba,\\[1em] (a - b)^2= a^2 - 2ab + b^2 \)

Hence, proved.

3. Proof of \(\pmb{(a + b)(a - b) = a^2 - b^2}\)

\((a + b)(a - b) = a(a - b) + b(a - b) \\[1em] \text{Upon evaluating,}\\[1em] (a + b)(a - b) = a^2 - ab + ba - b^2 \\[1em] \text{Upon simplifying},\\[1em] (a + b)(a - b) = a^2 - b^2\)

Hence, proved.

4. Proof of \(\pmb{(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3}\)

\((a + b)^3 = (a + b)(a + b)(a + b) \\[1em] \text{Upon evaluating,}\\[1em] (a + b)^3= (a^2 + 2ab + b^2)(a + b) \\[1em] (a + b)^3= a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3 \\[1em] \text{Upon simplifying,}\\[1em] (a + b)^3= a^3 + 3a^2b + 3ab^2 + b^3 \)

Hence, proved.

5. Proof of \(\pmb{(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3}\)

\( (a - b)^3 = (a - b)(a - b)(a - b) \\[1em] \text{Upon evaluating,}\\[1em] (a - b)^3 = (a^2 - 2ab + b^2)(a - b) \\[1em] (a - b)^3 = a^3 - a^2b - 2a^2b + 2ab^2 + ab^2 - b^3 \\[1em] \text{Upon simplifying,}\\[1em] (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)

Hence, proved.

6. Proof of \(\pmb{a^3 + b^3 = (a + b)(a^2 - ab + b^2)}\)

\((a + b)(a^2 - ab + b^2) = a(a^2 - ab + b^2) + b(a^2 - ab + b^2) \\[1em] \text{Upon evaluating,}\\[1em] (a + b)(a^2 - ab + b^2)= a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3 \\[1em] \text{Upon simplifying,}\\[1em] (a + b)(a^2 - ab + b^2)= a^3 + b^3 \)

Hence, proved.

7. Proof of \(\pmb{a^3 - b^3 = (a - b)(a^2 + ab + b^2)}\)

\((a - b)(a^2 + ab + b^2) = a(a^2 + ab + b^2) - b(a^2 + ab + b^2) \\[1em] \text{Upon evaluating,}\\[1em] (a - b)(a^2 + ab + b^2)= a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 \\[1em] \text{Upon simplifying,}\\[1em] (a - b)(a^2 + ab + b^2)= a^3 - b^3 \)

Hence, proved.

Verifying an algebraic identity simply means that both sides of the equation are equal for all values of the variables. We have some systematic ways to verify an equation, depending on the expression types. Some of the common methods are given as;

1. Expansion method: The expansion method is one of the most common and standard method to verify an algebraic identity. It can be done by following the given steps.

Step 1: Expanding both the sides of the expression completely.
Step 2: Simplifying each side by combining their like terms. 
Step 3: If both sides are identical in the end, the identity is verified. 

2. Substitution method: We use this method to check whether the given identity is true or not. It can be done by following the given steps.

Step 1: Substitute some numerical values in place of the variables.
Step 2: Simplify LHS and RHS separately.
Step 3: If we get the same result for LHS and RHS, the identity is true for those values.

3. Factorization method: In this method, both the sides are factored and then compared. 

Step 1: Factorize both LHS and RHS into simpler algebraic factors
Step 2: If the factorized forms are the same, then the identity is verified. 

4. Using algebraic manipulation and simplification: One can either multiply or divide the expression by a non-zero number on both sides, or we can add or subtract any term on both the sides. 

To solve complex equations, students use algebraic identities. So let’s learn the importance of algebraic identities for students.
 

• Algebraic identities are used to solve complex equations by breaking down the complex equations. 
   
• For factorization of polynomial equations, students use algebraic identities 
   
• Helps in understanding the logic behind algebraic identities and understand the basics of algebraic concepts.

By learning algebraic identities, students can easily solve the complex equation easier. It is important for students to learn and understand their identities. To make the process easier, let’s discuss some tips and tricks.

• Memorizing basic identities: By memorizing the basic algebraic identities, students can easily understand the concept of algebraic identities. For example, using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we can easily memorize the below-mentioned identities:  
  
  \((a - b)^2 = a^2 - 2ab + b^2\)
  \(a^2 - b^2 = (a + b)(a - b)\)
  \((x + a)(x + b) = x^2 + (a + b)x + ab\)
   
• Understanding the pattern: By understanding the pattern of the equations, students can match it with known identities. For instance, \(x^2 - 16\) can be written as \(x^2 - 4^2\).
   
• Regular revision: Since algebraic identities form the base of advanced mathematics, revising them regularly helps retain the formulas and apply them instantly during problem-solving.
   
• Solve step-by-step: While applying identities, students should expand or simplify step-by-step to avoid mistakes. Breaking down the process ensures accuracy in solving complex problems.
   
• Practice with numbers first: Students can substitute small numbers in place of variables to test whether an identity holds true. This helps in understanding the structure before applying it in algebraic equations.
   
• Start with concepts, not formulas: Teachers must start teaching concepts rather than formulas. They must help students understand why we use identities.
   
• Connect with real-life: Parents and teachers must show how identities help us simplify real problems, such as mentally finding squares and cubes. Instead of using the expansion technique for big numbers, we can simplify easily using these identities.
   
• Encourage pattern discovery: Teachers should encourage the students to discover the pattern of an equation. Instead of just giving all the formulas at once, teachers should let the students learn the patterns themselves.
   
• Practice with purpose: Parents should encourage their children to practice with a purpose in algebraic identities. Instead of making them do repetitive drills, please give them a variety of drills. Ask them to verify identities by expanding, substituting numbers, or simplifying algebraic expressions using the identities.

Now we learn about algebraic identities, so let's see how we use it in our real world. In the real world, we use it in the fields of engineering, physics, computer science, and so on.

• Construction: Algebra is used in construction to calculate elevations and design structures. 
   
• Budgeting: When calculating the exchange rates and interest rates, we use algebra to make it easier.
   
• Computer programming: Algebra is used to design algorithms, solve complex problems, and optimize codes.
   
• Physics: Algebraic identities are used in equations of motion, energy calculations, and force analysis to simplify and solve problems.
   
• Engineering: Engineers use identities for stress analysis, bridge design, and calculating load distributions.
   
• Factorization techniques: The equation is broken down into simpler equations to make them easier to solve.
  
  For example, \(x^3 - 8\)
  Using the difference in cube \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
  
  \(x^3 - 8\) can be written as \(x^3 - 2^3\)
  So, \(x^3 - 2^3 = (x - 2)(x^2 + 2x + 2^2)\)
   
• Expanding expressions: Expanding the expression using identities can make the calculation direct and easier.  
  
  For example, expanding \((2 + 3)^3\) using binomial expansion,
  \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
  
  That is, \((2 + 3)^2 = 2^3 + 3 × (2^2) × 3 + 3 × 2 × 3^2 + 93\)
  \((2 + 3)^2 = 8 + 36 + 54 + 27 = 125.\)