What is the HCF of `8(x^5 - x^3+x) and 28("x"^(6)+1)` ?

Options

A. `4(x^4 - x^2 +1)`
B. `x^3 - x+4x^2`
C. `x^3 - x+3x^2`
D. None of these

Answer

To find the HCF (Highest Common Factor) of the expressions \(8(x^5 - x^3 + x)\) and \(28(x^6 + 1)\), we can follow these steps:

### Step 1: Factor out the coefficients and common terms
First, we can factor out the coefficients from both expressions.

1. For \(8(x^5 - x^3 + x)\), we can factor out \(8\):
\[ 8(x^5 - x^3 + x) = 8 \cdot (x^5 - x^3 + x) \]

2. For \(28(x^6 + 1)\), we can factor out \(28\):
\[ 28(x^6 + 1) = 28 \cdot (x^6 + 1) \]

### Step 2: Identify the common factors
Next, we need to identify the common factors in the coefficients \(8\) and \(28\).

- The prime factorization of \(8\) is \(2^3\).
- The prime factorization of \(28\) is \(2^2 \cdot 7\).

The highest common factor of the coefficients \(8\) and \(28\) is \(4\) (which is \(2^2\)).

### Step 3: Factor the polynomial expressions
Now, we will factor the polynomial expressions:

1. For \(x^5 - x^3 + x\):
\[ x^5 - x^3 + x = x(x^4 - x^2 + 1) \]

2. For \(x^6 + 1\), we can use the sum of cubes formula:
\[ x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1) \]
Here, we set \(A = x^2\) and \(B = 1\) in the formula \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\).

### Step 4: Write the expressions with their factors
Now we can rewrite the original expressions with their factors:

- \(8(x^5 - x^3 + x) = 8 \cdot x(x^4 - x^2 + 1)\)
- \(28(x^6 + 1) = 28 \cdot (x^2 + 1)(x^4 - x^2 + 1)\)

### Step 5: Identify the common polynomial factor
The common polynomial factor between the two expressions is \(x^4 - x^2 + 1\).

### Step 6: Combine the common factors
Now we can combine the common factors to find the HCF:
\[ \text{HCF} = 4(x^4 - x^2 + 1) \]

### Final Answer
Thus, the HCF of \(8(x^5 - x^3 + x)\) and \(28(x^6 + 1)\) is:
\[ \text{HCF} = 4(x^4 - x^2 + 1) \]

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