Math Problem Statement

Factor by grouping.

x cubedx3minus−x squaredx2plus+44xminus−4

Solution

We need to factor the polynomial x3x2+4x4x^3 - x^2 + 4x - 4 by grouping.

Step 1: Group the terms

Group the terms in pairs to factor them more easily:

(x3x2)+(4x4)(x^3 - x^2) + (4x - 4)

Step 2: Factor out the greatest common factor (GCF) from each group

Factor out the GCF from each group:

x2(x1)+4(x1)x^2(x - 1) + 4(x - 1)

Step 3: Factor out the common binomial

Now, we can see that both terms have a common factor of (x1)(x - 1). Factor this out:

(x1)(x2+4)(x - 1)(x^2 + 4)

Conclusion

The polynomial x3x2+4x4x^3 - x^2 + 4x - 4 factors as (x1)(x2+4)(x - 1)(x^2 + 4).

Would you like any further explanation, or do you have any other questions?


Related Questions:

  1. How would you solve x3x2+4x4=0x^3 - x^2 + 4x - 4 = 0 using the factors?
  2. What are the roots of the equation x1=0x - 1 = 0?
  3. Can x2+4x^2 + 4 be factored further over the real numbers?
  4. What does factoring by grouping mean, and when is it used?
  5. How do you check if your factorization is correct?

Tip: When factoring by grouping, always look for common factors in pairs of terms. This method is particularly useful when dealing with four-term polynomials.

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Math Problem Analysis

Mathematical Concepts

Polynomial factoring
Grouping terms
Greatest common factor (GCF)
Binomial factorization

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12