https://file.aiquickdraw.com/mathbot/file/1728572642615jtupnsg6.jpg

The expression provided is \( 2x^4 - 2x^3 + x^2 - x \) and the instruction "tekijöihin" translates from Finnish to "factorize."

Let's proceed with factoring the given polynomial:

### Step-by-Step Factorization:
1. **Identify common factors:**
   The greatest common factor (GCF) of all the terms is \( x \), so we can factor out \( x \) from each term:
   \[
   x(2x^3 - 2x^2 + x - 1)
   \]

2. **Factor the cubic expression:**
   Now, we need to factor the cubic polynomial \( 2x^3 - 2x^2 + x - 1 \). We can try factoring by grouping:
   \[
   2x^3 - 2x^2 + x - 1 = (2x^3 - 2x^2) + (x - 1)
   \]
   Factor out \( 2x^2 \) from the first two terms:
   \[
   = 2x^2(x - 1) + 1(x - 1)
   \]

3. **Combine like factors:**
   Now we have a common factor of \( (x - 1) \), so we can factor that out:
   \[
   = (2x^2 + 1)(x - 1)
   \]

4. **Final factored form:**
   So the fully factored expression is:
   \[
   x(2x^2 + 1)(x - 1)
   \]

This is the factorized form of the given polynomial.

Would you like further details or clarification on any part of this solution? Here are five related questions for you:

1. How do you determine the greatest common factor (GCF) when factoring polynomials?
2. What is factoring by grouping, and when is it used in polynomial factorization?
3. Can the expression \( 2x^2 + 1 \) be factored further over the real numbers?
4. Why do we group terms in a cubic polynomial for factorization?
5. How can synthetic division or the Rational Root Theorem help in factoring cubic polynomials?

**Tip:** Always start by factoring out the greatest common factor before attempting more complex factorization methods.

Factor the polynomial expression 2x^4 - 2x^3 + x^2 - x.

Learn how to factor the polynomial 2x^4 - 2x^3 + x^2 - x with a step-by-step solution. This problem involves factoring by grouping and identifying the greatest common factor. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation