Problem 19 - Elementary Algebra

Question

Factor completely. $$ -3 x 2+75 $$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

Look for common factors in the terms \(-3x^2\) and \(75\). Both terms are divisible by \(-3\). Extract \(-3\) as a common factor:\[-3(x^2 - 25)\].

Step 2: Recognize the Difference of Squares

The expression \(x^2 - 25\) is a difference of squares, which fits the formula \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a^2 = x^2\) and \(b^2 = 25\).

Step 3: Apply the Difference of Squares Formula

Factor \(x^2 - 25\) using the difference of squares formula:\((x + 5)(x - 5)\).

Step 4: Combine the Factors

Combine the factors found in the previous steps. The completely factored form is:\[-3(x + 5)(x - 5)\].

Key Concepts

Difference of Squares

The difference of squares is a special type of polynomial expression that takes the form \(a^2 - b^2\). This type of expression can be factored using a specific formula.
The formula for factoring a difference of squares is:

• \(a^2 - b^2 = (a + b)(a - b)\)

This works because when expanding \((a + b)(a - b)\), the middle terms cancel each other out, which results in \(a^2 - b^2\).
Let's apply this to an example from the exercise: the expression is \(x^2 - 25\).
- Here, \(x^2\) is \(a^2\) where \(a = x\), and \(25\) is \(b^2\) where \(b = 5\).- Using the difference of squares formula, we can factor this as \((x + 5)(x - 5)\).Understanding the difference of squares formula allows us to quickly transform complex quadratic expressions into simpler linear factors.

Common Factor Extraction

Common factor extraction is a fundamental technique in polynomial factoring, allowing us to simplify polynomial expressions by factoring out the greatest common factor (GCF). To identify the GCF in a polynomial, look at each term and determine the largest factor that divides all terms.In our exercise, the polynomial \(-3x^2 + 75\) contains two terms:

• \(-3x^2\)
• \(75\)

Both these terms can be divided by \(-3\), making it the greatest common factor.
By factoring \(-3\) out of each term, we simplify the polynomial to:- \(-3(x^2 - 25)\)This step is crucial as it not only simplifies the expression but also lays the groundwork for further factoring, such as recognizing the difference of squares within the remaining expression.
Common factor extraction makes the process of factoring more manageable and organized.

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables and coefficients, constructed from the sum of more than one monomial. Polynomials can vary in degree and complexity, often requiring different techniques for factoring.
In the given exercise, we have a polynomial with terms:- \(-3x^2\) - and \(+75\).This degree-two polynomial demonstrates a straightforward process that combines multiple factoring methods.
Key points to remember include:

• Identifying common factors across terms.
• Recognizing special patterns, like difference of squares, that allow for quick factoring.

When approaching polynomial expressions, always perform a check for the greatest common factor first. From there, look for any special factoring identities such as the difference of squares.
Polynomials can initially appear daunting, but by breaking them down into their simpler components through these techniques, they become easier to manage and understand.