# Notable Products

*See also:*binomials, trinomials*Ref.*polynomials

*Notable products* refer to specific products of powers, binomials, and trinomials, which result from multiplying expressions with particular patterns. Such products lead to simplified and recognizable forms frequently occurring in mathematical equations. Notable products have a significant role in algebraic simplification, polynomial factorization, and problem-solving in mathematics. These products are essential for solving equations, as they allow for more straightforward calculations and provide a basis for identifying common mathematical patterns.

## Square of a binomial

The square of a binomial \((a+b)^2\) or \((a-b^2)\) expands to the sum of the square of the first term and twice the product of the first and second terms plus the square of the second term. This expansion is commonly referred to as the FOIL method, where F stands for First terms, O stands for Outer terms, I stands for Inner terms, and L stands for Last terms.

To determine the square of a binomial expression \((a+b)^2\), one may consider \(a\) as the first term and \(b\) as the second term, and using the FOIL methods, we obtain:

- \(F \quad a \cdot a = a^2\)
- \(O \quad a \cdot b = ab\)
- \(I \quad b \cdot a = ab\)
- \(L \quad b \cdot b = b^2\)

Adding the terms, we obtain:

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

To determine the square of a binomial expression \((a-b)^2\), following the same method, we obtain:

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

## Difference of two squares

A factorization formula exists for the difference of two squares, represented as \(a^2-b^2\). This formula expresses the difference of two squares as the product of the sum of the two squares \((a+b)\) and the difference of the two squares \((a-b)\). The factorization is expressed as:

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

Indeed, proceeding backwards, we obtain:

\begin{align*} (a+b)(a-b) &= a(a – b) + b(a-b)\\[0.5em] &= a^2 – ab+ab-b^2 \\[0.5em] &= a^2-b^2 \end{align*}

## Cube of a binomial

The cube of a binomial is defined as the algebraic sum or difference, elevated to the cube, of two monomials, \(a\) and \(b\). Finding the cube of a binomial involves multiplying the binomial by itself three times, resulting in a trinomial. The trinomial obtained from the cube of a binomial can be simplified using the binomial theorem, which provides a formula for expanding the powers of a binomial \((a+b)^n\). In short, you have:

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

## Sum and difference of two cubes

The addition or difference of two cubes constitutes a distinct case in algebra, wherein two terms are raised to the power of three and then added or subtracted. The formula of expansion is:

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

## Sum of nth powers

The sum of nth powers expressed as \(a^n+b^n\) is a mathematical problem that involves finding the sum of two terms raised to the nth power, where \(n\) is a positive integer. This problem has no general formula for any value of \(n\), and the calculation depends on whether n is even or odd.

If \(n\) is even, then \(a^n+b^n\) can be factored into a sum that can be expressed in terms of lower powers of \(a\) and \(b\):

\[ a^n + b^n = (a + b)(a^{n-1} – a^{n-2}b + a^{n-3}b^2 – \ldots – ab^{n-2} + b^{n-1}) \]

The formula is similar to the sum of cubes when \( n = 3\). It expresses \( a^n + b^n \) as a product of \(a + b \) and a sum of terms involving alternating powers of \( a \) and \( b \). It facilitates the representation of a summation of two numbers raised to the power of \(n\) as a product of the sum of the two numbers and a series of terms that alternate between the powers of each number. The resulting expression embodies a polynomial of degree \(n\) consisting of a series of terms involving alternating powers of the two numbers.

When \(n\) is an even positive integer, the sum of nth powers, \(a^n + b^n\), doesnâ€™t have a simple factorization formula like the one for the sum of cubes for odd powers. Nevertheless, there are some patterns and properties that can be useful. For example, if \(a\) and \(b\) are equal, the sum becomes:

\[ a^n + a^n = 2a^n \]

In conclusion, notable products offer essential techniques that make solving equations and simplifying expressions easier. In addition to simplifying complex expressions, these formulas have fundamental applications in algebra, geometry, and physics.