The HCF of `x^4 -11x^2 +10, x^2 – 5x+4` and `x^3 – 3x^2 + 3x-1 `is

Options

A. `x+1 `
B. `x-4`
C. `x+2 `
D. `x-1`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(x^4 - 11x^2 + 10\), \(x^2 - 5x + 4\), and \(x^3 - 3x^2 + 3x - 1\), we will follow these steps:

### Step 1: Factor each polynomial

1. **Factor \(x^4 - 11x^2 + 10\)**:
- We can rewrite it as \(x^4 + 0x^3 - 11x^2 + 0x + 10\).
- We look for two numbers that multiply to \(10\) (the constant term) and add to \(-11\) (the coefficient of \(x^2\)).
- The numbers \(-10\) and \(-1\) work since \(-10 \times -1 = 10\) and \(-10 + -1 = -11\).
- We can rewrite the polynomial as:
\[ x^4 - 10x^2 - x^2 + 10 = (x^2 - 10)(x^2 - 1) \]
- Further factor \(x^2 - 1\) using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
- Thus, the complete factorization is:
\[ x^4 - 11x^2 + 10 = (x^2 - 10)(x - 1)(x + 1) \]

2. **Factor \(x^2 - 5x + 4\)**:
- We look for two numbers that multiply to \(4\) and add to \(-5\).
- The numbers \(-4\) and \(-1\) work since \(-4 \times -1 = 4\) and \(-4 + -1 = -5\).
- Therefore, we can factor it as:
\[ x^2 - 5x + 4 = (x - 4)(x - 1) \]

3. **Factor \(x^3 - 3x^2 + 3x - 1\)**:
- We can use synthetic division or factor by grouping.
- Grouping gives us:
\[ (x^3 - 3x^2) + (3x - 1) = x^2(x - 3) + 1(3x - 1) \]
- This does not seem to simplify easily, so we can check for roots. Testing \(x = 1\):
\[ 1^3 - 3(1^2) + 3(1) - 1 = 0 \]
- Thus, \(x - 1\) is a factor. We can divide \(x^3 - 3x^2 + 3x - 1\) by \(x - 1\) using synthetic division:
\[ x^3 - 3x^2 + 3x - 1 = (x - 1)(x^2 - 2x + 1) = (x - 1)(x - 1)^2 = (x - 1)^3 \]

### Step 2: Identify common factors

Now we have the factorizations:
- \(x^4 - 11x^2 + 10 = (x^2 - 10)(x - 1)(x + 1)\)
- \(x^2 - 5x + 4 = (x - 4)(x - 1)\)
- \(x^3 - 3x^2 + 3x - 1 = (x - 1)^3\)

### Step 3: Find the HCF

The common factor in all three polynomials is \(x - 1\). The powers of \(x - 1\) are:
- In the first polynomial: \(1\)
- In the second polynomial: \(1\)
- In the third polynomial: \(3\)

The HCF is the lowest power of the common factor:
\[ \text{HCF} = (x - 1)^1 = x - 1 \]

### Final Answer:
The HCF of the given polynomials is \(x - 1\).

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