Polynomial Identities

• Polynomial identities are equations that are true for all values of the variables.

Difference of Squares

• The difference of squares identity is a useful tool in algebra.
• For any two terms a and b: a² - b² = (a + b)(a - b)
• Factoring a difference of squares involves recognizing the pattern and applying the formula in reverse.
• Expanding the right side proves the identity: (a + b)(a - b) = a² - ab + ba - b² = a² - b²
• Example: x² - 9 can be factored as (x + 3)(x - 3).
• Example: 4y² - 25 can be factored as (2y + 5)(2y - 5).
• When expanding, the identity helps to quickly multiply expressions of the form (a + b)(a - b).
• Example: (x + 4)(x - 4) expands to x² - 16.
• Example: (3y + 2)(3y - 2) expands to 9y² - 4.
• The key is to identify the terms that are being squared.

Binomial Expansion

• Binomial expansion involves raising a binomial (an expression with two terms) to a power.
• A binomial is an algebraic expression of two terms, such as (a + b) or (x - y).
• The binomial theorem provides a formula for expanding binomials raised to a power.
• The binomial theorem: (a + b)ⁿ = Σ [n! / (k!(n-k)!)] * a^(n-k) * b^k, for k = 0 to n
• For practical expansion, Pascal's triangle or the distributive property can be used for smaller powers.
• Using the distributive property (also known as FOIL - First, Outer, Inner, Last) is common for binomials to the power of 2.
• (a + b)² = (a + b)(a + b) = a² + 2ab + b²
• (a - b)² = (a - b)(a - b) = a² - 2ab + b²
• Expanding (a + b)² results in a² + 2ab + b².
• Expanding (a - b)² results in a² - 2ab + b².
• Factoring involves recognizing these patterns and expressing the quadratic as a squared binomial.
• Example: x² + 6x + 9 can be factored as (x + 3)².
• Example: y² - 4y + 4 can be factored as (y - 2)².
• Expanding (a + b)³ results in a³ + 3a²b + 3ab² + b³.
• Expanding (a - b)³ results in a³ - 3a²b + 3ab² - b³.
• Pascal's triangle provides the coefficients for binomial expansion.
• Each row starts and ends with 1, and the numbers in between are the sum of the two numbers above.
• Row 0: 1 (for (a + b)⁰)
• Row 1: 1 1 (for (a + b)¹)
• Row 2: 1 2 1 (for (a + b)²)
• Row 3: 1 3 3 1 (for (a + b)³)
• Row 4: 1 4 6 4 1 (for (a + b)⁴)
• Example: Expand (x + 2)².
• Using the formula: (x + 2)² = x² + 2(x)(2) + 2² = x² + 4x + 4.
• Example: Expand (y - 3)².
• Using the formula: (y - 3)² = y² - 2(y)(3) + 3² = y² - 6y + 9.
• When expanding, pay attention to the signs, especially when dealing with (a - b)ⁿ.