91Ó°ÊÓ

When dealing with binomials raised to a power, such as \( (3y+4)^2 \), we utilize a technique known as binomial expansion. A binomial is simply an algebraic expression containing two terms, in our case, \( 3y \) and \( 4 \).

The formula used for expansion is \( (a+b)^2 = a^2 + 2ab + b^2 \), where \( a \) and \( b \) are the two terms of the binomial.

• Step 1: Square the first term, which is \( (3y)^2 = 9y^2 \).
• Step 2: Multiply the two terms together and then double the result: \( 2 \times (3y) \times 4 = 24y \).
• Step 3: Square the second term: \( 4^2 = 16 \).

Now, add all these results together to get the expanded form: \( 9y^2 + 24y + 16 \). This process simplifies a squared binomial into a polynomial.

The Distributive Property is a fundamental algebraic principle used when multiplying a single term across a polynomial. In our exercise, we apply the Distributive Property to \( 4y \) and the polynomial \( 9y^2 + 24y + 16 \).

This property states that when multiplying a term by a sum, distribute that term to each addend of the sum. In simpler words:

• Multiply: Apply \( 4y \) to each term of the polynomial separately.
• For example, \( 4y \times 9y^2 \) becomes \( 36y^3 \).
• Then, \( 4y \times 24y \) gives \( 96y^2 \).
• Lastly, \( 4y \times 16 \) results in \( 64y \).

Combine the results to finalize the expression. The Distributive Property allows for complex expressions to be expanded, making it easier for further calculations.

Algebraic expressions form the backbone of algebra with variables, coefficients, and operations. When you look at the expression \( 4y(3y+4)(3y+4) \), several key components form the structure:

• Variables: Symbols like \( y \) represent unknown quantities that can change.
• Coefficients: These are numbers that multiply the variables, such as \( 4 \) in \( 4y \).
• Constants: Fixed values like \( 4 \) in the binomial \( 3y+4 \).

In this exercise, we manipulated our expression using algebraic rules to achieve simplification. Working with these elements, especially through multiplication and expansion, requires careful attention to the rules governing algebra.By observing how terms interact and following algebraic principles, complex expressions can be simplified into manageable forms like \( 36y^3 + 96y^2 + 64y \). This process not only aids in calculations but also deepens the understanding of how algebraic expressions function.