FACTORISE THIS FUNCTION USING THE U-SUBSISTUTION METHOD. X^3 -X^2 -2X + 4

To factor the function \( f(x) = x^3 - x^2 - 2x + 4 \) using the **u-substitution method**, we'll need to apply some trial and error along with polynomial division or synthetic division. However, the function seems to benefit more from factoring by grouping. Let me guide you step by step:

### Step 1: Group the terms
We group the terms to make factoring easier:
\[
x^3 - x^2 - 2x + 4 = (x^3 - x^2) - (2x - 4)
\]

### Step 2: Factor each group
Factor out the greatest common factor (GCF) from each group:
\[
= x^2(x - 1) - 2(x - 1)
\]

### Step 3: Factor out the common binomial
Notice that \( (x - 1) \) is common in both terms. We can factor that out:
\[
= (x - 1)(x^2 - 2)
\]

### Step 4: Simplify
Now, check if the quadratic term \( x^2 - 2 \) can be factored further. It’s a difference of squares:
\[
x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})
\]

### Final Factored Form:
Thus, the completely factored form is:
\[
f(x) = (x - 1)(x - \sqrt{2})(x + \sqrt{2})
\]

Would you like further details on any step?

---

### Related Questions:
1. How do you factor polynomials by grouping?
2. What is synthetic division and when is it useful in factoring?
3. Can all cubic functions be factored by the u-substitution method?
4. What are difference of squares and how are they factored?
5. How do you recognize the best factoring technique for a given polynomial?

### Tip:
When factoring polynomials, always look for the greatest common factor or opportunities to group terms.

Learn how to factor the cubic polynomial X^3 - X^2 - 2X + 4 using the u-substitution method, with an emphasis on factoring by grouping and applying the difference of squares. This step-by-step guide explores key algebraic concepts and factoring techniques. Suitable for high school 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