The HCF of `x^5 +2x^4 + x^3` and `x^7 – x^5` is

Options

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

Answer

To find the HCF (Highest Common Factor) of the polynomials \( x^5 + 2x^4 + x^3 \) and \( x^7 - x^5 \), we will follow these steps:

### Step 1: Factor the first polynomial \( x^5 + 2x^4 + x^3 \)

1. **Identify common factors**: Notice that \( x^3 \) is a common factor in all terms.
2. **Factor out \( x^3 \)**:
\[ x^5 + 2x^4 + x^3 = x^3(x^2 + 2x + 1) \]
3. **Factor the quadratic**: The expression \( x^2 + 2x + 1 \) can be factored as:
\[ x^2 + 2x + 1 = (x + 1)^2 \]
4. **Combine the factors**:
\[ x^5 + 2x^4 + x^3 = x^3(x + 1)^2 \]

### Step 2: Factor the second polynomial \( x^7 - x^5 \)

1. **Identify common factors**: Here, \( x^5 \) is a common factor.
2. **Factor out \( x^5 \)**:
\[ x^7 - x^5 = x^5(x^2 - 1) \]
3. **Factor the difference of squares**: The expression \( x^2 - 1 \) can be factored as:
\[ x^2 - 1 = (x - 1)(x + 1) \]
4. **Combine the factors**:
\[ x^7 - x^5 = x^5(x - 1)(x + 1) \]

### Step 3: Identify the HCF

Now we have factored both polynomials:
- \( x^5 + 2x^4 + x^3 = x^3(x + 1)^2 \)
- \( x^7 - x^5 = x^5(x - 1)(x + 1) \)

To find the HCF, we look for the common factors:
- The common factor is \( x + 1 \) (from both factorizations).
- The lowest power of \( x \) is \( x^3 \) from the first polynomial.

Thus, the HCF is:
\[ \text{HCF} = x^3(x + 1) \]

### Final Answer
The HCF of \( x^5 + 2x^4 + x^3 \) and \( x^7 - x^5 \) is \( x^3(x + 1) \).

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