Problem 34 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (3 x+4)^{2} $$

Step-by-Step Solution

Answer

The simplified result is \(9x^2 + 24x + 16\).

Step 1: Write the Expression

The given expression is \((3x + 4)^{2}\). This means you need to multiply \((3x + 4)\) by itself.

Step 2: Apply the Binomial Theorem

Use the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 3x\) and \(b = 4\).

Step 3: Calculate \(a^2\)

Calculate \((3x)^2 = 9x^2\).

Step 4: Calculate \(2ab\)

Multiply \(2\) by \(3x\) and \(4\): \(2 \times 3x \times 4 = 24x\).

Step 5: Calculate \(b^2\)

Calculate \(4^2 = 16\).

Step 6: Combine All Parts

Combine the results: \(9x^2 + 24x + 16\).

Step 7: Present the Simplified Expression

The simplified expression is \(9x^2 + 24x + 16\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making math problems easier to read and work with. Think of it like tidying up your room - everything in its right place and easy to find. When you're working with expressions such as \((3x + 4)^2\), the goal is to express it in a more manageable form.

In the provided exercise, this meant expanding the squared term into a standard polynomial form. The key reasons for simplifying include:

• Making calculations easier and faster.
• Solving equations or inequalities efficiently.
• Making graphing simpler.

When you simplify an expression, ensure that you correctly combine like terms, distribute properly, and follow mathematical rules like the order of operations.

Algebraic Operations

Algebraic operations are the actions you perform on algebraic expressions. These include addition, subtraction, multiplication, and division. The binomial theorem is a crucial tool in algebraic operations, specifically when dealing with squares or expansions of binomials.

For the exercise \((3x + 4)^2\), applying these operations helps to break down the problem:

• Multiplication is used when you apply the binomial theorem: \((a + b)^2 = a^2 + 2ab + b^2\).
• Addition combines the different parts of the binomial expansion into a unified expression.
• Understanding how to perform each operation aids in reducing errors and solving problems accurately.

The better you grasp algebraic operations, the smoother the simplification process will be.

Polynomials

Polynomials are expressions involving variables raised to whole number exponents. They can include constants and coefficients as seen in the expression \(9x^2 + 24x + 16\). This expression is a polynomial of degree 2, also known as a quadratic polynomial.

Understanding polynomials is important because they form the basis of many algebraic concepts:

• They are foundational in equations and functions studied across math courses.
• Recognizing the structure of polynomials (terms, coefficients, exponents) is key to manipulating and solving them.
• Knowing how to expand and simplify polynomials, as shown in the step-by-step solution, is vital for various applications including calculus and scientific computing.

Mastering polynomials enables you to handle more complex mathematical problems efficiently.