Problem 83 - College Algebra Essentials

Question

In Exercises 83–90, perform the indicated operation or operations. $$(3 x+4 y)^{2}-(3 x-4 y)^{2}$$

Step-by-Step Solution

Answer

The solution to the exercise is \( 0 \)

Step 1: Expanding the Squares

Expand both squares using formula \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\). For the first square \(a = 3x\) and \(b = 4y\), for the second one \(a = 3x\) and \(b = -4y\). This results in:\[(3 x+4 y)^{2} = (3x)^{2} + 2*(3x)*(4y) + (4y)^{2}\]\[(3 x-4 y)^{2} = (3x)^{2} - 2*(3x)*(-4y) + (-4y)^{2}\]

Step 2: Simplifying the Squares

Simplify both of the above expressions:\[(3 x+4 y)^{2} = 9x^{2} + 24xy + 16y^{2}\]\[(3 x-4 y)^{2} = 9x^{2} + 24xy + 16y^{2}\]

Step 3: Subtracting the Squares

Subtract \((3 x-4 y)^2\) from \((3 x+4 y)^2\) which results in \(0\).

Step 4: Final Simplification

Simplify the expression to get the final answer: \( 0 \)

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operators (like addition or multiplication). They don't have an equality sign like equations do. One way to make sense of algebraic expressions is to identify different parts of them:

• Variables: They represent unknown values. In our exercise, the variables are \(x\) and \(y\).
• Coefficients: Numbers that multiply the variables, like 3 in \(3x\).
• Constants: Numbers alone without any variables, although this exercise doesn't include them explicitly.

The expression \((3x + 4y)^2\) and \((3x - 4y)^2\) in our original exercise are both algebraic because they feature numbers, operations, and variables. Understanding each component helps us manipulate and expand the expression as needed.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in an extended form. In our given exercise, we're using the expansion of a binomial to show the difference of squares. A binomial is an algebraic expression that contains two different terms, like \((3x + 4y)\). When we expand the square of a binomial, we use well-known patterns:

• The square of a sum: \[(a+b)^2 = a^2 + 2ab + b^2\] This formula helps expand \((3x + 4y)^2\).
• The square of a difference: \[(a-b)^2 = a^2 - 2ab + b^2\] This allows us to expand \((3x - 4y)^2\).

Both formulas break down a complex expression into simpler individual terms that can be multiplied and simplified. This is what we did in Step 1 and Step 2 of the solution, making it easier to subtract and simplify them later.

Difference of Squares

The difference of squares is a neat little algebraic trick. It comes from the fact that any algebraic expression like \(a^2 - b^2\) can be rewritten into two binomials multiplied together: \[(a + b)(a - b) = a^2 - b^2\] In practical terms, it means that something like \(x^2 - y^2\) can always be decomposed into \((x+y)(x-y)\). In our exercise, after expanding and simplifying the polynomials \((3x+4y)^2\) and \((3x-4y)^2\), we are left with a subtraction that leverages this identity, though it doesn't appear at first glance. When expanded, both expressions cancel each other out entirely, showcasing a simplified example of how the difference of squares pattern leads directly to a resolution of zero. Understanding this principle can significantly simplify certain algebraic operations.