The expression to factorize is \( 3a^4 + 2a^2b^2 - b^4 \).

Let's factorize it step by step:

1. **Rewrite the expression** in a form that makes it easier to factor:
   \[
   3a^4 + 2a^2b^2 - b^4 = (3a^4 + 2a^2b^2 - b^4)
   \]

2. **Recognize this as a quadratic in terms of \( a^2 \)**. Let \( x = a^2 \), so the expression becomes:
   \[
   3x^2 + 2xb^2 - b^4
   \]

3. Now, **factorize the quadratic** expression:
   \[
   3x^2 + 2xb^2 - b^4 = (3x - b^2)(x + b^2)
   \]

4. Substitute \( x = a^2 \) back into the factors:
   \[
   (3a^2 - b^2)(a^2 + b^2)
   \]

So, the factorized form of \( 3a^4 + 2a^2b^2 - b^4 \) is:
\[
(3a^2 - b^2)(a^2 + b^2)
\]

Would you like more details or have any questions about this factorization?

Here are 5 related questions for further exploration:
1. How do you factorize a quadratic expression?
2. What are the common factorization techniques for polynomials?
3. How do you recognize if a polynomial can be factored into simpler terms?
4. What is the difference between factorizing over real numbers versus complex numbers?
5. Can the expression \( a^2 + b^2 \) be further factorized over complex numbers?

**Tip:** When factorizing expressions, it's helpful to look for patterns, such as recognizing quadratics or special identities like difference of squares.

Learn how to factorize the polynomial expression 3a^4 + 2a^2b^2 - b^4 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial factorization and the difference of squares formula. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation