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When you come across an expression like \( (4a + 7b)^2 \), you'll need to understand the Binomial Square Formula to solve it. This formula helps you simplify expressions where a binomial (a term made up of two parts) is being squared.

The Binomial Square Formula is given by: \[ (x + y)^2 = x^2 + 2xy + y^2 \] Let's break this down:

• \( x^2 \) means we square the first term.
• \( 2xy \) means we multiply both terms together, then multiply the result by 2.
• \( y^2 \) means we square the second term.

Here, the binomial is \( 4a + 7b \), so:

• \( x = 4a \) and \( y = 7b \)
• Applying the formula: \[ (4a + 7b)^2 = (4a)^2 + 2(4a)(7b) + (7b)^2 \]

By expanding each term, you simplify the expression.

Algebraic expressions are like sentences of math that use numbers, variables, and operators (like +, -, *, and /). They help us describe mathematical scenarios. For example, \( 4a + 7b \) is an algebraic expression.

When you multiply and simplify algebraic expressions, you're broadening the use of basic operations:

• Combining like terms (terms with the same variables raised to the same power).
• Understanding and using operations such as addition, subtraction, multiplication, and division.

Let’s see these steps in action with our binomial:
1. Separately calculate the square of each term:
\((4a)^2 = 16a^2\)
\( (7b)^2 = 49b^2 \)

2. Calculate the middle term:
\ (2 \times 4a \times 7b = 56ab) \

3. Combine all the results:
\ 16a^2 + 56ab + 49b^2 \ This process illustrates the interaction of algebraic expressions in polynomial equations.

Polynomial expansion is a technique used to expand expressions that have multiple terms, like binomials. The aim is to simplify them into a sum of simpler terms. This makes complex expressions easier to manage.

When we expanded \( (4a + 7b)^2 \), we turned a binomial into a polynomial by using the Binomial Square Formula. This is a fundamental aspect of polynomial expansion:

The general relationship in polynomial expansion for any \( (x + y)^n \) is based on the Binomial Theorem. This however, is derived from simpler expansions:

• Recognize the pattern: Using the formula \( (x + y)^2 = x^2 + 2xy + y^2 \)
• Apply to the expression, turning it from a compact form into an expanded form.

In our specific case, \( (4a + 7b)^2 \) expanded to: \[ 16a^2 + 56ab + 49b^2 \] This highlights how polynomial expansion provides a structured method to transform and simplify algebraic expressions.