91Ó°ÊÓ

Algebra is a significant branch of mathematics that deals with symbols and the rules for manipulating these symbols. The symbols used in algebra are usually numbers and variables. In the exercise above, the expression

• (x - 6)^2

is an algebraic expression where 'x' is a variable and '6' is a constant.
In algebraic terms, the square of a binomial expression like

• (x - 6)

means multiplying the binomial by itself, i.e.,

• (x - 6) * (x - 6)

This step follows the rules and formulas that algebraic expressions are governed by.
Algebra allows us to take expressions filled with variables and constants and find common patterns and formulas to simplify, solve, or expand them. It's like a toolkit for dissecting and reassembling complicated mathematical statements into simpler forms.

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The exercise given involves squaring the polynomial

• (x - 6)

which is a polynomial consisting of two terms:

• 'x' (the variable) and
• '-6' (the constant term).

When squared, the expression becomes a trinomial, having three terms as a result of the expansion.
The general product of a polynomial with itself uses the distributive property to expand, which is fundamental to understanding polynomial arithmetic.
The expanded form

is a polynomial in which each term represents:

• x^2 - the square of the first term
• -12x - twice the product of the two terms
• +36 - the square of the second term

Binomial expansion refers to expressing a binomial expression raised to a power in expanded form. In the given exercise,

• (x - 6)^2

is expanded using the square of a binomial formula.
The square of a binomial formula

• (a - b)^2 = a^2 - 2ab + b^2

helps us quickly expand expressions without writing out the full multiplication and prevents common errors made during distribution.
**Steps for Binomial Expansion**:

• Deduce values: Identify 'a' and 'b' from the binomial
• Expand using formula: Substitute 'a' and 'b' into (a - b)^2 = a^2 - 2ab + b^2
• Simplify: Combine and simplify terms to write the final polynomial expression

Recognizing and applying this formula ensures that you expand binomials efficiently, reducing the effort needed and minimizing mistakes.