Problem 47 - Algebra A Combined Function

Question

Factor each completely. $$ (x+2 y)^{2}-9 $$

Step-by-Step Solution

Answer

\((x + 2y - 3)(x + 2y + 3)\)

Step 1: Recognize the Difference of Two Squares

The expression \((x+2y)^2 - 9\) can be viewed as the difference of two squares, where \(a^2 = (x+2y)^2\) and \(b^2 = 9\). This matches the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x+2y\) and \(b = 3\).

Step 2: Apply the Difference of Squares Formula

According to the formula, substitute \(a = x + 2y\) and \(b = 3\) into \((a - b)(a + b)\). This gives us:\((x + 2y - 3)(x + 2y + 3)\).

Step 3: Verify the Factorization

To ensure the factorization is correct, expand \((x + 2y - 3)(x + 2y + 3)\) and verify that it results in the original expression. Expanding yields:\[(x + 2y)^2 + 3(x + 2y) - 3(x + 2y) - 9 = (x + 2y)^2 - 9\], confirming the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra that involves expressions of the form \(a^2 - b^2\). Such expressions can be factored into the form \((a-b)(a+b)\).
This formula is quite handy for simplifying polynomials and solving equations. Here, the expression given was \((x+2y)^2 - 9\).
By identifying \(a = x + 2y\) and \(b = 3\), we observe that both \(a^2\) and \(b^2\) are perfect squares. Thus, it fits the pattern of a difference of squares.

- Recognize the expression as two squares being subtracted.
- Use the formula \((a-b)(a+b)\) to factor directly and efficiently.

This technique reduces the complexity of the original expression by transforming it into a product of two simpler binomials.

Polynomial Expressions

Polynomial expressions comprise terms that are made up of coefficients, variables, and exponents combined using addition, subtraction, and multiplication.
The given expression \((x+2y)^2 - 9\) is a polynomial because it involves the square of a binomial and a constant.
Understanding how to manipulate polynomial expressions, including recognizing their patterns, is crucial in algebra.

- Recognizing patterns such as squares, cubes, and their differences or sums can simplify the factorization process.
- Identifying the correct formula to apply, like the difference of squares in this situation, is key to simplifying the polynomial.

Comprehending the structure of polynomial expressions allows for easier expansion and simplification, leading to the correct factorization solutions.

Expansion and Verification

The process of expansion and verification ensures that the factorization performed is accurate. After applying the formula for the difference of squares in the exercise, the result was \((x+2y-3)(x+2y+3)\).
To verify, we expand these binomials to check if they lead back to the original expression.
Expansion involves distributing each term in the first binomial over each term in the second binomial. Let's see how this works:
- Expand: \((x+2y-3)(x+2y+3)\) becomes \((x+2y)^2 + 3(x+2y) - 3(x+2y) - 9\).
- Simplify: Notice the \(+3(x+2y) - 3(x+2y)\) terms cancel each other out, leaving \((x+2y)^2 - 9\).

- This confirms the factorization is correct.- Always compare your expanded result back to the original expression for verification.

Verification reinforces understanding of algebraic manipulations and correctness of calculations.