Algebraic Identities¶

Algebraic identities are standard mathematical formulas that are true for all values of the variables involved. These identities simplify complex algebraic expressions and are widely used in factorization, solving equations, and expanding expressions. Here’s a summary of key algebraic identities:

1. Square of a Sum¶

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

Example: \( (x + 3)^2 = x^2 + 6x + 9 \)

2. Square of a Difference¶

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

Example: \( (x - 4)^2 = x^2 - 8x + 16 \)

3. Product of a Sum and Difference¶

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

This is the difference of squares identity.

Example: \( (x + 5)(x - 5) = x^2 - 25 \)

4. Cube of a Sum¶

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

Example: \( (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \)

5. Cube of a Difference¶

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

Example: \( (x - 1)^3 = x^3 - 3x^2 + 3x - 1 \)

6. Sum of Cubes¶

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

Example: \( x^3 + 8 = (x + 2)(x^2 - 2x + 4) \)

7. Difference of Cubes¶

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

Example: \( x^3 - 27 = (x - 3)(x^2 + 3x + 9) \)

8. Absolute Value Properties¶

Multiplication:

\[ |a \cdot b| = |a| \cdot |b| \]

Addition (Triangle Inequality):

\[ |a + b| \leq |a| + |b| \]

8. General Binomial Theorem (for any power \(n\))¶

The binomial theorem expands powers of sums:

\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

where \(\binom{n}{k}\) is the binomial coefficient.

Example for \(n = 2\): \( (a + b)^2 = a^2 + 2ab + b^2 \)

These algebraic identities are fundamental tools for expanding, factoring, and simplifying algebraic expressions in higher mathematics.