Problem 89 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \((2 x+y-3)(2 x+y+3)\)

Step-by-Step Solution

Answer

The simplified form is \( 4x^2 + 4xy + y^2 - 9 \).

Step 1: Apply the Difference of Squares Formula

Recognize the expression \( (2x + y - 3)(2x + y + 3) \) as a difference of squares, which takes the form \( (a - b)(a + b) = a^2 - b^2 \). In this case, let:- \( a = 2x + y \) - \( b = 3 \). So, the operation becomes:\( ((2x + y)^2 - 3^2) \).

Step 2: Square the Binomial

Square the binomial \( (2x + y) \), which involves expanding \( (2x + y)^2 \).Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):\[ (2x + y)^2 = (2x)^2 + 2(2x)(y) + y^2 = 4x^2 + 4xy + y^2 \].

Step 3: Square the Constant

Calculate the square of the constant:\\[ 3^2 = 9 \].

Step 4: Combine the Results

Substitute the squared terms back into the difference of squares equation:\[ (2x + y)^2 - 3^2 = 4x^2 + 4xy + y^2 - 9 \].

Step 5: Simplify the Result

Check the expression \( 4x^2 + 4xy + y^2 - 9 \) for any like terms, which there aren't any.The simplified form of the expression is:\( 4x^2 + 4xy + y^2 - 9 \).

Key Concepts

Binomial Expansion

The process of expanding a binomial is commonly used in algebra to break down expressions of the form \((a + b)^2\). The goal is to rewrite it as a sum of terms. In our original exercise, we had to expand \((2x + y)^2\). Expanding this correctly requires remembering the formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

Here, \(a\) is \(2x\) and \(b\) is \(y\). When we apply the formula:

• \((2x + y)^2 = (2x)^2 + 2(2x)(y) + y^2\)
• \((2x)^2 = 4x^2\)
• \(2(2x)(y) = 4xy\)
• \(y^2 = y^2\)

So, the expanded form becomes \(4x^2 + 4xy + y^2\). It's essential to perform these calculations step-by-step to avoid errors and correctly handle each component of the binomial.

Algebraic Simplification

Simplifying algebraic expressions is about reducing them to their simplest form while maintaining their original value. In the original exercise, we used one type of simplification called the difference of squares, followed by combining like terms. First, we acknowledge the presence of a difference of squares:

• This was seen in \((2x + y - 3)(2x + y + 3)\) in the form \((a-b)(a+b) = a^2 - b^2\).

After recognizing and applying the formula, the expression transformed into:

• \((2x + y)^2 - 3^2\)

Following the expansion of the binomial and calculating the square of the constant:

• \((2x+y)^2 = 4x^2 + 4xy + y^2\)
• \(3^2 = 9\)
• The operation became \(4x^2 + 4xy + y^2 - 9\).

The simplified expression forms naturally from the calculated components. Ensuring no like terms were overlooked, the expression retained the form \(4x^2 + 4xy + y^2 - 9\). Understanding simplification helps in solving equations efficiently and effectively.

Polynomial Expressions

Polynomial expressions consist of variables, coefficients, and exponents all combined by addition or subtraction. Consider our example expression of the form \(4x^2 + 4xy + y^2 - 9\), which is a polynomial:

• Each term involves powers of variables. \(4x^2\) and \(y^2\) are quadratic terms.
• \(4xy\) represents a mixed or cross term involving both \(x\) and \(y\).
• The \(-9\) is a constant term, simply a number without any variable attached.

Polynomial expressions are a key part of algebra, providing the foundation for further study in calculus, mathematical modeling, and more.
They can represent real-world phenomena or theoretical constructs, making their simplification and manipulation essential skills. Identifying terms as mentioned above helps decipher and manage these expressions effectively in operations such as addition, subtraction, multiplication, and division.