Problem 75 - Algebra 2

Question

Factor each polynomial. $$ 3 x^{2}-75 $$

Step-by-Step Solution

Answer

The factored form is \(3(x - 5)(x + 5)\).

Step 1: Identify the Common Factor

First, look for any common factors in both terms of the polynomial. Here, both terms have a factor of 3. So, factor 3 out of the entire expression:\[ 3x^{2} - 75 = 3(x^{2} - 25) \]

Step 2: Recognize the Difference of Squares

Observe that the expression inside the parentheses \(x^2 - 25\) can be rewritten as a difference of squares. Recall that a difference of squares has the form \(a^2 - b^2\), which factors into \((a-b)(a+b)\). Here, recognize that \(x^2\) is \(x^2\) and \(25\) is \(5^2\).

Step 3: Apply the Difference of Squares Formula

Use the difference of squares formula to factor \(x^2 - 25\):\[ x^2 - 25 = (x - 5)(x + 5) \]Now substitute this back into the expression:\[ 3(x^2 - 25) = 3((x - 5)(x + 5)) \]

Step 4: Finalize the Factorization

The factorization is complete with the common factor 3 factored out and the expression \(x^2 - 25\) factored using the difference of squares:\[ 3(x - 5)(x + 5) \]

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental element of polynomial factorization. It occurs when a polynomial can be rewritten in the form of the difference between two perfect squares, like this: \( a^2 - b^2 \). The special property of a difference of squares is that it can be factored into the product of two binomials: \((a - b)(a + b)\).
This technique is incredibly useful because it simplifies expressions and aids in solving quadratic equations. In our example, \(x^2 - 25\) is a difference of squares since \(x^2\) is a square \((of\ x)\), and \(25\) is also a square \((5^2)\).

• Recognize forms like \(a^2 - b^2\).
• Factor them into \((a - b)(a + b)\).
• Simplify complex expressions.

Mastering this technique is essential for efficient algebraic manipulation, and it's a stepping stone to more advanced algebra concepts.

Common Factor

Discovering a common factor is an important first step in the factorization of any polynomial. A common factor is a term that can evenly divide each term of the polynomial without leaving a remainder. Recognizing and extracting common factors simplifies the polynomial and opens the way for further factorization.
In our exercise, the original polynomial is \(3x^2 - 75\). Here, we search for a term that appears in both \(3x^2\) and \(75\). Both terms are divisible by \(3\), so we pull out this common factor:

• Look for common integers or variables in all terms.
• Extract common factors.
• Rewrite the polynomial in a simpler form: \(3(x^2 - 25)\).

Factorizing by common factors is crucial as it prepares complex expressions for further simplification, like recognizing the difference of squares, as seen in our next steps.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators that follow the rules of algebra to form a coherent mathematical statement. Factorizing these expressions is an essential skill which helps solve equations and understand more complex mathematical concepts.
One key task when working with algebraic expressions is to identify patterns or special forms that facilitate easy manipulation of the expressions. In the exercise we tackled, we started with simplifying the original algebraic expression \(3x^2 - 75\) by finding a common factor and then recognizing the difference of squares.

• Identify special forms like difference of squares.
• Use algebraic identities to simplify expressions.
• Combine factorization methods for efficient calculation.

Developing expertise with these expressions is fundamental in achieving a deeper understanding of mathematics, as it allows students to break down and solve problems effectively.