Problem 94 - Algebra A Combined Function

Question

Square where indicated. Simplify if possible. \((5 x+2 y)^{2}\)

Step-by-Step Solution

Answer

The expression \((5x + 2y)^2\) simplifies to \(25x^2 + 20xy + 4y^2\).

Step 1: Understand the Expression

The expression given is \((5x + 2y)^2\). This is a perfect square trinomial, which means it is of the form \((a + b)^2\). Our goal is to expand and simplify it.

Step 2: Apply the Formula for a Perfect Square

The formula for a perfect square \((a + b)^2\) is \(a^2 + 2ab + b^2\). Here, \(a = 5x\) and \(b = 2y\). We will substitute these values into the formula.

Step 3: Calculate Each Term

First, calculate \(a^2\): \((5x)^2 = 25x^2\). Next, calculate \(2ab\): \(2 \cdot 5x \cdot 2y = 20xy\). Finally, calculate \(b^2\): \((2y)^2 = 4y^2\).

Step 4: Construct the Expanded Expression

Using the terms calculated in Step 3, construct the expanded expression: \(25x^2 + 20xy + 4y^2\).

Step 5: Simplify If Possible

The expression \(25x^2 + 20xy + 4y^2\) is already simplified as it contains no like terms that can be combined further. Hence, this is the final expression.

Key Concepts

Expanding Binomials

Expanding binomials is a fundamental concept in algebra that involves taking an expression of the form \((a + b)^n\) and writing it as a sum of terms. This process is especially useful in simplifying expressions and solving equations. To understand expansion, we typically use the distributive property of multiplication over addition. For a binomial squared, like \((5x + 2y)^2\), you apply the formula \((a + b)^2 = a^2 + 2ab + b^2\). In our exercise:

• Set \(a\) as \(5x\)
• Set \(b\) as \(2y\)
• Perform the expansions: \((5x)^2 + 2(5x)(2y) + (2y)^2\)

This expands the expression into separate terms.Breaking it down this way helps us understand how each part of a binomial contributes to the final expanded form.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and performing any other operations necessary to present the expression in its simplest form. In the context of our original exercise, the expanded binomial \(25x^2 + 20xy + 4y^2\) has already been simplified. No further combination of like terms can occur because:

• Each term is unique
• No terms share the same variables raised to the same powers

The process ensures that expressions are easier to work with in further operations, whether that means graphing, plugging values in, or solving equations. It's crucial to always check if an expression can be simplified further by looking for like terms or common factors.

Mathematical Formulas

Mathematical formulas are predefined rules or expressions that show relationships between different mathematical entities. In algebra, formulas act as shortcuts that help solve problems efficiently. A common formula used in expanding binomials is the perfect square formula, which states: \((a + b)^2 = a^2 + 2ab + b^2\). In the provided exercise, this formula allowed us to expand the binomial without having to manually multiply \((5x + 2y)(5x + 2y)\) in its entirety. By substituting \(a\) and \(b\) with \(5x\) and \(2y\), respectively, students can quickly arrive at an expansion without extra steps. Formulas like these save time and reduce errors, making them indispensable tools in mathematics.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, incorporating operations such as addition, subtraction, and multiplication. Polynomials come in many forms, defined by the number of terms and degree they contain. In our context, a trinomial is part of this family, composed of three terms after expansion.The expanded result \(25x^2 + 20xy + 4y^2\) exemplifies a polynomial expression.

• It contains three distinct terms: \(25x^2\), \(20xy\), and \(4y^2\)
• Each term consists of coefficients and variables raised to varying powers

Understanding polynomial expressions allows students to perform further operations, such as factoring, evaluating, and graphing. Recognizing patterns in polynomials helps in solving more complex problems efficiently and effectively.