Problem 51 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((3 x+4)^{2}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Formula

Recognize that the expression \((3x + 4)^2\) can be expanded using the special product formula for the square of a binomial. The formula is \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 3x\) and \(b = 4\).

Step 2: Apply the Formula

Substitute \(a = 3x\) and \(b = 4\) into the formula: \((3x + 4)^2 = (3x)^2 + 2(3x)(4) + 4^2\).

Step 3: Calculate Each Term

Calculate each term separately: \((3x)^2 = 9x^2\).For the middle term, calculate \(2(3x)(4) = 24x\).Finally, calculate \(4^2 = 16\).

Step 4: Combine the Terms

Combine all the calculated terms: the expression becomes \(9x^2 + 24x + 16\).

Step 5: Simplified Expression

The simplified form of the original expression \((3x + 4)^2\) is \(9x^2 + 24x + 16\).

Key Concepts

Expanding Binomials

In algebra, expanding binomials is an essential skill that requires you to understand special product formulas. Binomials are expressions with two terms, such as \((a + b)\). When you raise a binomial to a power, such as squaring \((3x + 4)^2\), you use special formulas to make it easier. One of the most popular special product formulas is the square of a binomial, given by: \[ (a + b)^2 = a^2 + 2ab + b^2 \] For the expression \((3x + 4)^2\), you start by identifying \(a\) and \(b\) from your binomial: \(a = 3x\) and \(b = 4\). Then, substitute these values into the formula.

• First term: \((3x)^2\) becomes \(9x^2\).
• Second term: \(2(3x)(4)\) simplifies to \(24x\).
• Third term: \(4^2\) becomes \(16\).

By expanding such expressions using the binomial formula, you break down complex operations into simple calculations. This helps in accurately multiplying the terms without errors.

Polynomial Expressions

Polynomials are expressions involving multiple terms combined using addition, subtraction, and multiplication. The terms can include constants, variables, and whole number exponents. When expanding binomials like \((3x + 4)^2\), the result is a polynomial expression: \(9x^2 + 24x + 16\). Understanding the structure of polynomials is crucial. The expanded expression has:

• A squared term: \(9x^2\)
• A linear term: \(24x\)
• A constant term: \(16\)

These components are categorized by their degree, which is the highest power of the variable. In this case, the degree is 2, originating from \(9x^2\), making it a quadratic polynomial. By recognizing these components, you can manipulate and solve polynomials more readily as you build on algebraic skills.

Algebraic Simplification

Simplifying algebraic expressions is about combining like terms and reducing expressions to their simplest form. When you expand a binomial like \((3x + 4)^2\), you get \(9x^2 + 24x + 16\). This is already simplified, meaning no further combination of like terms is possible. Simplification involves a few steps:

• Perform all arithmetic operations like powers and multiplications.
• Combine terms that have the same variables and exponents.
• Ensure no common factors exist between terms.

The result is a cleaner, more manageable expression that can be easily interpreted and utilized in problem-solving. Simplifying expressions makes it easy to identify coefficients, terms, and ultimately, the solution to algebraic problems. Understanding this process helps in further mathematical tasks, ensuring accuracy and clarity in your calculations.