Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((3 x-2)^{2}\)

Step-by-Step Solution

Answer

The product of \((3x-2)^{2}\) is \(9x^{2}-12x+4\).

Step 1: Identify a and b

In our exercise, the binomial is \((3x-2)^{2}\). Here \(a=3x\) and \(b=2\).

Step 2: Substitution into the formula

We substitute our \(a\) and \(b\) into the binomial theorem formula \((a-b)^{2}=a^{2}-2ab+b^{2}\). Therefore, we get \((3x-2)^{2}=(3x)^{2}-2*3x*2+2^{2}\).

Step 3: Simplification

Solving above equation we get \(9x^{2}-12x+4\). This is our final answer.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to represent numbers in general forms and solve equations in a structured way.

In this exercise, algebra is used to apply the binomial theorem, which helps simplify the expression \((3x - 2)^2\). Understanding these operations is essential as it provides a foundation for more advanced math topics.

To use algebra effectively, follow some basic rules:

• Identify the components of the expression (in this case, 3x and -2).
• Apply the correct formula for expansion, here it's the square of a binomial.
• Substitute the values, and simplify step by step.

By mastering these basics, you'll be able to handle more complex operations with confidence.

Quadratic Expressions

When you expand a binomial like \((3x - 2)^2\), you get a quadratic expression. Quadratic expressions involve terms where the highest exponent of the variable is 2.

The general form of a quadratic expression is \(ax^2 + bx + c\). In our case, expanding \((3x - 2)^2\) gives us a quadratic expression:

• \(9x^2\) representing the \(a\) term \((a = 9)\)
• \(-12x\) representing the \(b\) term \((b = -12)\)
• \(+4\) representing the \(c\) term \((c = 4)\)

Each of these terms plays an important role in shaping the curve of the quadratic. The \(x^2\) term defines the curve's openness, while the linear \(x\) term and constant term \(c\) adjust the curve's position and shape on the graph.

Polynomials

A polynomial is a mathematical expression that consists of variables (also called indeterminates) and coefficients. These expressions are combined using operations like addition, subtraction, and multiplication.

In this example, \(9x^2 - 12x + 4\) is a polynomial, specifically a quadratic polynomial, as its highest degree term is 2.

When analyzing polynomials, you should:

• Identify the terms separated by + or -.
• Determine the degree of the polynomial, which is the highest power of the variable.
• Simplify or combine like terms using algebra.

Understanding the structure of polynomials allows you to perform operations like factoring, expansion, and solving polynomial equations systematically. Recognizing the hierarchy in terms (from the highest degree to the constant term) is crucial for effectively working with these expressions.