Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (2 x+3 y)^{2} $$

Step-by-Step Solution

Answer

The simplified expression for \((2x+3y)^2\) is \(4x^2 + 12xy + 9y^2\).

Step 1: Understand the problem

We are given the expression \((2x+3y)^2\). Our task is to expand and simplify this expression.

Step 2: Apply the binomial square rule

To expand and simplify this expression, we use the binomial square rule, which states that the square of a binomial expression \((a+b)^2\) is given by: \[ (a+b)^2 = a^2 + 2ab + b^2 \] Here, \(a=2x\) and \(b=3y\).

Step 3: Expand the expression

Using the binomial square rule, we expand our expression: \[ (2x+3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 \]

Step 4: Simplify the expanded expression

Now, we will simplify the expression by performing the calculations: \[ (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2 \] So, the simplified expression is: \[ (2x+3y)^2 = 4x^2 + 12xy + 9y^2 \]

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is key to grasping fundamental mathematical concepts. An algebraic expression comprises variables, numbers, and operations that describe a mathematical relationship. In its simplest form, an expression can be just a term with a variable, like \(2x\), or a constant, like \(3\).

Complex expressions include multiple terms combined through arithmetic operations like addition, subtraction, multiplication, or division. In the given exercise, \(2x + 3y\)^2 is an example of a binomial expression, which consists of two terms: \(2x\) and \(3y\). Each term consists of variables with coefficients. Algebraic expressions serve as the building blocks for forming equations and inequalities that model real-world scenarios.

Learning to work with algebraic expressions involves understanding how to manipulate and simplify them while maintaining their equivalence. This is fundamental in solving equations and in calculus later.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form, making it easier to work with and understand. Polynomials are expressions that consist of variables raised to whole number powers, combined using addition, subtraction, and multiplication.

When simplifying polynomials, the goal is to combine like terms and eliminate unnecessary parentheses. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression \(4x^2 + 12xy + 9y^2\), no further simplification by combining terms is possible because each term is distinct.

To simplify complex expressions, identify and combine all like terms. Use arithmetic and associative properties, distribute multiplication over addition or subtraction, and follow the order of operations. For the given expression, simplifying was straightforward after expanding using the binomial theorem. It resulted in the polynomial \(4x^2 + 12xy + 9y^2\), which can't be simplified further.

Binomial Theorem

The Binomial Theorem provides a powerful formula for expanding expressions raised to a power. It tells us how to express \(a + b\)^n as a sum involving terms of the form \(a^k b^{n-k}\). When the exponent \(n\) equals 2, we have the special case used in the exercise: the binomial square rule.

The theorem states \(a + b)^2 = a^2 + 2ab + b^2\). Here, \(a\) and \(b\) represent the terms of the binomial. Recognizing patterns in binomial expansions is crucial for simplifying expressions. By applying this rule, we expanded \(2x + 3y\)^2 into \(4x^2 + 12xy + 9y^2\).

The Binomial Theorem extends to larger powers and can be applied to solve problems in algebra, combinatorics, and probability. It benefits from Pascal's Triangle in its expression of coefficients, with each coefficient corresponding to the terms of the expanded binomial. Mastery of this theorem opens doors to deeper mathematical concepts and problem-solving techniques.