Which of the following is equivalent to the expression `x^4-x^3-x^2`?

Options

A. `x(x^2-x-1)`
B. `x(x-x^2-x^3)`
C. `x(x^3-x^2)`
D. `x^2(x^2-x-1)`

Answer

To solve the expression \( x^4 - x^3 - x^2 \) and find its equivalent form, we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ x^4 - x^3 - x^2 \]

### Step 2: Factor out the common term
We notice that each term in the expression has a common factor of \( x^2 \). Therefore, we can factor \( x^2 \) out of the expression:
\[ x^4 - x^3 - x^2 = x^2(x^2 - x - 1) \]

### Step 3: Write the factored form
After factoring out \( x^2 \), we are left with:
\[ x^2(x^2 - x - 1) \]

### Step 4: Identify the equivalent expression
Now we compare our factored expression \( x^2(x^2 - x - 1) \) with the options provided:
1. \( x(x^2 - x - 1) \)
2. \( x(x - x^2 - x^3) \)
3. \( x(x^3 - x^2) \)
4. \( x^2(x^2 - x - 1) \)

We can see that our factored form matches option 4:
\[ x^2(x^2 - x - 1) \]

### Conclusion
Thus, the expression \( x^4 - x^3 - x^2 \) is equivalent to option 4:
\[ \text{Option 4: } x^2(x^2 - x - 1) \]

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