If (x+2) is one of the factors of X³+x²-4x-4 what are the other factors

To find the other factors of \( x^3 + x^2 - 4x - 4 \) given that \( (x+2) \) is a factor, we can use polynomial division.

### Step-by-Step Solution:

1. **Set up the division**: We divide \( x^3 + x^2 - 4x - 4 \) by \( x + 2 \).

2. **Perform long division**:
   - Divide \( x^3 \) by \( x \), which gives \( x^2 \).
   - Multiply \( x^2 \) by \( (x + 2) \), resulting in \( x^3 + 2x^2 \).
   - Subtract \( (x^3 + 2x^2) \) from \( x^3 + x^2 - 4x - 4 \), leaving \( -x^2 - 4x \).

   - Divide \( -x^2 \) by \( x \), which gives \( -x \).
   - Multiply \( -x \) by \( (x + 2) \), resulting in \( -x^2 - 2x \).
   - Subtract \( (-x^2 - 2x) \) from \( -x^2 - 4x \), leaving \( -2x - 4 \).

   - Divide \( -2x \) by \( x \), which gives \( -2 \).
   - Multiply \( -2 \) by \( (x + 2) \), resulting in \( -2x - 4 \).
   - Subtract \( (-2x - 4) \) from \( -2x - 4 \), leaving a remainder of 0.

3. **Conclusion**: The quotient is \( x^2 - x - 2 \).

Now, factor \( x^2 - x - 2 \):
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Thus, the complete factorization of \( x^3 + x^2 - 4x - 4 \) is:
\[
(x + 2)(x - 2)(x + 1)
\]

### Final Answer:
The other factors are \( (x - 2) \) and \( (x + 1) \).

Would you like more details or explanations on any part?

### Related Questions:
1. How do you perform synthetic division for polynomials?
2. Can all cubic polynomials be factored using this method?
3. How can you solve for the roots of a cubic equation directly?
4. What happens when a polynomial has complex factors?
5. How do you factor a cubic polynomial that doesn't have obvious rational roots?

### Tip:
Always check for possible factoring shortcuts like grouping or synthetic division when dealing with polynomials!

If (x+2) is one of the factors of x³+x²-4x-4, what are the other factors?

Discover how to factor the cubic polynomial x³+x²-4x-4 step-by-step using polynomial division. Given one factor (x+2), find the other factors and learn key algebraic techniques including polynomial division and quadratic factorization. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation