Problem 39 - A Graphical Approach to College Algebra

Question

Find each product. $$(2 m+3)(2 m-3)$$

Step-by-Step Solution

Answer

The product is \(4m^2 - 9\).

Step 1: Identify the Expression Type

The expression \[(2m + 3)(2m - 3)\]is in the form of a difference of squares. The general form of a difference of squares is \[(a + b)(a - b) = a^2 - b^2.\]Here, \(a = 2m\) and \(b = 3\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, we substitute \(a = 2m\) and \(b = 3\) into the expression. Thus, \[(2m + 3)(2m - 3) = (2m)^2 - 3^2.\]

Step 3: Calculate the Squares

Now, we need to calculate \[(2m)^2\] and \[3^2:\] - \((2m)^2 = (2)^2 imes (m)^2 = 4m^2\)- \(3^2 = 9\)

Step 4: Simplify the Expression

Substitute the calculated values back into the expression:\[4m^2 - 9.\]

Step 5: Write the Final Product

Thus, the final simplified product of the expression \[(2m + 3)(2m - 3)\] is \[4m^2 - 9.\]

Key Concepts

Factoring

Factoring is an essential process in algebra where you write an expression as a product of its components, called factors. It's like breaking down a number into its multipliers. For a quadratic expression, factoring can make equations easier to solve or simplify. For example, expressions like

• \((x^2 - 4)\) can be factored into \((x + 2)(x - 2)\), which is a difference of squares.
• Expressions not in simple difference of squares form might be factored with other methods like grouping or using special factorization formulas.

In factoring quadratic expressions, recognizing patterns, like a difference of squares, can simplify the process. Using the formula \((a+b)(a-b) = a^2 - b^2\), you identify suitable terms in your expression, making it easier to factor.

Quadratic Expressions

Quadratic expressions are polynomials where the highest power of the variable is 2, usually in the form \(ax^2 + bx + c\). These expressions can often be factored to find their roots or simplify them for analysis.Some common techniques to factor them include:

• Identifying them as a product of binomials, sometimes appearing as a difference of squares, like in the exercise \((2m + 3)(2m - 3)\) which simplifies to \(4m^2 - 9\).
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) can solve equations derived from these expressions, though factoring is faster when applicable.
• Observing coefficients and constants can help identify factorizable patterns or arrangements.

Spotting these characteristics can lead to quicker and more accurate problem-solving strategies.

Polynomials

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Often used in algebraic equations, polynomials are foundational in algebra and calculus. They come in various forms:

• Monomials, such as \(3x\), a single term polynomials.
• Binomials, such as \(x^2 + 4\), two terms.
• Polynomials of higher degree, which may include quadratic \(ax^2 + bx + c\), cubic \(ax^3 + bx^2 + cx + d\), and so forth.

Polynomials can be manipulated in equations via addition, subtraction, multiplication, and factoring to solve equations. Recognizing their forms and patterns, like the difference of squares, allows for efficient problem-solving and simplification as seen with the expressions \((2m+3)(2m-3)\). Learning to identify and work with these patterns enhances algebraic fluency.