Algebraic Identities
2026-02-28 23:48 Diff
https://brightchamps.com/en-us/math/algebra/algebraic-identities

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.