Problem 15 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((a+3)^{2}\)

Step-by-Step Solution

Answer

The simplest form of \((a+3)^2\) is \(a^2 + 6a + 9\).

Step 1: Identify the Expression to Expand

The given expression is \[(a+3)^2\]This indicates that the bracketed term \(a + 3\) is to be squared, which means multiplying \(a + 3\) by itself.

Step 2: Apply the Squaring Formula

The square of a binomial \((x+y)^2\) is expanded using the formula:\[(x+y)^2 = x^2 + 2xy + y^2\]For our problem, let \(x = a\) and \(y = 3\) to apply this formula.

Step 3: Substitute Values into Formula

Using the formula from the previous step, substitute \(x = a\) and \(y = 3\):\[(a + 3)^{2} = a^2 + 2(a)(3) + 3^2\]

Step 4: Calculate Each Term

Now compute each part of the expanded expression:- \(a^2\) remains as \(a^2\).- \(2(a)(3) = 6a\).- \(3^2 = 9\).Substitute back into the expression:\[a^2 + 6a + 9\]

Step 5: Write the Final Expression

Combine all parts from Step 4 to give the expression in its simplest form:\[a^2 + 6a + 9\]

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like \(a\), \(x\), or \(y\)), and operators such as addition, subtraction, multiplication, and division. These expressions are fundamental components in algebra, playing a critical role in everything from solving equations to graphing functions. Understanding algebraic expressions involves:

• Identifying constants (specific numbers) and variables (symbols representing numbers).
• Recognizing the mathematical operations that connect these components.
• Learning the rules of operations, including the order of operations (PEMDAS/BODMAS).

Algebraic expressions come in many forms, from simple to complex, and can be manipulated to simplify and solve mathematical problems. In our exercise, we started with the algebraic expression \((a+3)^2\), which required expansion to simplify further.

Squaring a Binomial

Squaring a binomial refers to multiplying a two-term expression, or binomial, by itself. For the expression \((a + 3)^2\), this involves using the formula for the square of a binomial. This formula is:\[(x+y)^2 = x^2 + 2xy + y^2\]This formula is derived from the distributive property and helps to quickly expand the square of any binomial without multiplying each term individually. Applying this:

• Substitute \(x = a\) and \(y = 3\)
• Expand to \(a^2 + 2(a)(3) + 3^2\)

Understanding this concept is crucial because it saves time and reduces errors in the expansion process. Using this method ensures that students can efficiently tackle problems involving binomials. Following the step by step, we expanded \((a+3)^2\) into \(a^2 + 6a + 9\).

Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form by combining like terms, which are terms that have the same variable raised to the same power. For instance, in our expanded expression from earlier, \(a^2 + 6a + 9\), no further simplification is needed as there are no like terms that can be combined.To simplify a polynomial:

• Identify like terms, which can often be added together.
• Use distributive properties if necessary to combine or rearrange terms.
• Ensure that all terms are arranged in descending order of their powers (standard form).

Simplifying polynomials is essential as it provides the cleanest and most accurate representation of expressions, making it easier to evaluate or solve within broader mathematical operations. In our polynomial \(a^2 + 6a + 9\), it's fully simplified and expressed in the simplest polynomial form.