Completely Factorizing 3-3x^2 in Advanced Math

Factorizing expressions is a fundamental skill in algebra, particularly when dealing with polynomials. This article will guide you through the process of completely factorizing the expression 3 - 3x^2, which can be broken down into simpler steps through the application of various mathematical rules and techniques.

Factorizing 3 - 3x^2

The expression 3 - 3x^2 can initially appear complicated, but let's break it down step-by-step to make it more manageable.

Step 1: Identify the Common Factor

The first step is to look for a common factor in both terms. In this case, it's clear that both terms have a factor of 3. Factoring out the 3, we rewrite the expression as:

31 - x^2

Step 2: Apply the Difference of Squares Rule

Next, we need to recognize that the term within the parentheses, 1 - x^2, is a well-known form in algebra. Specifically, it's a difference of squares, which follows the pattern a^2 - b^2 (a - b)(a b). Here, a 1 and b x. Applying this rule, we can rewrite:

1 - x^2 (1 - x)(1 x)

This gives us the fully factorized form:

31 - x^2 3(1 - x)(1 x)

Final Answer

The completely factorized form of the expression 3 - 3x^2 is:

3(1 - x)(1 x)

Conclusion

Factoring expressions like 3 - 3x^2 can be a powerful tool in simplifying and solving more complex algebraic problems. By understanding the techniques and rules involved, such as identifying common factors and recognizing differences of squares, you can efficiently factorize even more intricate expressions.

Try It Yourself!

What have you tried so far and how far did you get? Hint 1: Look for a common factor in the terms. Hint 2: After factoring out, apply the difference of squares rule to further simplify the expression.

Challenge yourself to factorize similar expressions such as 5 - 5x^2 and see if you can apply the same techniques. Practice is key to mastering these algebraic skills.