Problem 55 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$y(4 x+3 y)(4 x-3 y)$$

Step-by-Step Solution

Answer

The result is \( 16x^2y - 9y^3 \).

Step 1: Distribute 'y' inside the expression

Since the expression is of the form \( y(a)(b) \), where \( a = (4x+3y) \) and \( b = (4x-3y) \), we first leave \( y \) aside to focus on the product of \( a \) and \( b \). We will only introduce 'y' after we've fully expanded and simplified \( (4x+3y)(4x-3y) \).

Step 2: Multiply the binomials

Apply the difference of squares formula to multiply the binomials: \((a+b)(a-b) = a^2 - b^2\). In our case, \(a = 4x\) and \(b = 3y\), so the result is \((4x)^2 - (3y)^2\).

Step 3: Expand the squares

Compute \((4x)^2\) and \((3y)^2\): \[(4x)^2 = 16x^2 \] \[(3y)^2 = 9y^2 \].Substituting back gives: \[ 16x^2 - 9y^2 \].

Step 4: Distribute 'y' across the result

Now distribute \( y \) into the expression \( 16x^2 - 9y^2 \) from Step 3:\[ y(16x^2 - 9y^2) = 16xy^2x - 9y^3 = 16x^2y - 9y^3 \].

Step 5: Simplify the expression

Write the final expression combining the terms with the distributed 'y':\[ 16x^2y - 9y^3 \].

Key Concepts

Binomial Multiplication

Binomial multiplication is a fundamental concept in algebra. It involves multiplying two binomials together, which are expressions consisting of two terms. For instance, in our exercise, we encounter

• The expression: \((4x + 3y)(4x - 3y)\).
• These terms have forms: \(a + b\) and \(a - b\).

When you multiply these binomials, it's helpful to identify patterns like the difference of squares. This pattern arises because any expression of the form \((a + b)(a - b)\) results in \(a^2 - b^2\).
Here, the terms were \(4x\) and \(3y\), leading us to \((4x)^2 - (3y)^2\). Recognizing these patterns can make complex multiplications much simpler.
Understanding binomial multiplication is essential as it forms the basis for more advanced topics in algebra, including factoring and solving quadratic equations.

Difference of Squares

The difference of squares is a specific pattern which is very useful in binomial multiplication. This concept states that

• When you multiply two binomials of the form \((a + b)(a - b)\), the result is \(a^2 - b^2\).

This pattern simplifies the multiplication process significantly and allows for quick calculation of complex algebraic expressions.
In our example, applying the difference of squares to \((4x)^2 - (3y)^2\) results in simplifying it to \(16x^2 - 9y^2\). Recognizing and understanding when to use this pattern is crucial for efficiently solving algebraic problems.
Mastering the difference of squares not only helps in multiplication but also is a powerful tool in factoring expressions, solving equations, and even calculus.

Distributive Property

The distributive property is a key principle in algebra that allows us to simplify expressions and solve equations systematically. It states that for any numbers or expressions \(a\), \(b\), and \(c\),

• \(a(b + c) = ab + ac\).

In this exercise, the distributive property is used twice:

• First, as the binomials \((4x + 3y)(4x - 3y)\) are multiplied using the difference of squares.
• Second, when we finally distribute \(y\) over the resulting expression \(16x^2 - 9y^2\), transforming it to \(16x^2y - 9y^3\).

Using the distributive property effectively can transform complex expressions into simpler ones, making them easier to handle. It is fundamental not only in algebra but also in various branches of mathematics and science.