Problem 63 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(36 x^{2}-100\)

Step-by-Step Solution

Answer

The expression factors to \(4(3x - 5)(3x + 5)\).

Step 1: Identify the Type of Expression

The given expression is a binomial: \(36x^2 - 100\). Notice that both terms are perfect squares: \(36x^2\) is \((6x)^2\) and 100 is \(10^2\). This suggests that the expression is a difference of squares.

Step 2: Apply Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). By comparing \(36x^2 - 100\) to \(a^2 - b^2\), we identify \(a = 6x\) and \(b = 10\).

Step 3: Factor Using the Formula

Substitute \(a\) and \(b\) into the difference of squares formula to factor the expression:\[36x^2 - 100 = (6x - 10)(6x + 10)\]

Step 4: Simplify the Factors, if Possible

Examine the factors \((6x - 10)\) and \((6x + 10)\) to see if further simplification is possible. Both can be simplified by factoring out the greatest common factor, which is 2:\[6x - 10 = 2(3x - 5)\]\[6x + 10 = 2(3x + 5)\]

Step 5: Rewrite Using Completely Factored Form

Replace each factor with the simplified version:\[36x^2 - 100 = 2(3x - 5) \times 2(3x + 5)\]Simplifying this further gives:\[36x^2 - 100 = 4(3x - 5)(3x + 5)\]

Key Concepts

Difference of Squares

The difference of squares is a special pattern in algebra that can make factoring easier. It refers to an expression of the form \(a^2 - b^2\). This type of expression stems from the difference between two perfect square numbers. The trick is that such expressions can be quickly factored using the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). This pattern is especially helpful because it breaks a more complex expression into two simpler binomial factors. These factors can often be simplified even further, which we'll talk about later on. When identifying a difference of squares, always check that both terms in the binomial expression are perfect squares and that they are being subtracted, not added.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler terms or factors that, when multiplied together, yield the original expression. It’s like reverse multiplying. A solid understanding of factoring is crucial in solving equations and simplifying expressions in algebra.

Some critical steps in factoring are:

• Identify common factors: Look for any common numbers, variables, or expressions that can be factored out.
• Apply special formulas: Such as the difference of squares, perfect square trinomials, or the sum and difference of cubes.
• Repeat if necessary: Sometimes, you'll need to factor more than once to simplify completely.

In the case of the expression \(36x^2 - 100\), we recognized it as a difference of squares with perfect square terms \((6x)^2\) and \(10^2\). After initially applying the formula, we further simplified using common factors. This two-step factoring ensured the expression was reduced to the simplest form possible.

Binomial Expression

A binomial expression is an algebraic expression that contains exactly two terms. These terms can be connected by either a plus or minus sign, making them either additive or subtractive. Binomials are one of the simplest forms of polynomial expressions in algebra, yet they play a vital role in more complicated algebraic manipulations.

Understanding binomials involves recognizing their structure and how they can be manipulated. For example:

• Check if the binomial is a candidate for special factoring techniques like difference of squares or perfect square trinomials.
• Assess whether the terms can be further simplified or factored individually, such as finding common factors.

In the original problem, the expression \(36x^2 - 100\) is a binomial because it consists of two terms. We used its binomial nature to apply the difference of squares technique, simplifying the structure and reducing it step by step to achieve the most straightforward representation.