HCF of expression `x^(4) + 3x^(2) - 4 and x^(4) - 4x^(2) + 3` will be

Options

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

Answer

To find the HCF (Highest Common Factor) of the expressions \( x^4 + 3x^2 - 4 \) and \( x^4 - 4x^2 + 3 \), we will factorize both expressions step by step.

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

We can rewrite the expression as:
\[ x^4 + 3x^2 - 4 = x^4 - 4 + 3x^2 \]

Next, we will group the terms:
\[ = (x^4 - 4) + 3x^2 \]

Now, we can factor \( x^4 - 4 \) using the difference of squares:
\[ x^4 - 4 = (x^2 - 2)(x^2 + 2) \]

So, we rewrite the expression:
\[ = (x^2 - 2)(x^2 + 2) + 3x^2 \]

Now, we will factor by grouping:
\[ = x^2(x^2 - 2) + 3(x^2 - 2) \]
\[ = (x^2 + 3)(x^2 - 2) \]

### Step 2: Factor the second expression \( x^4 - 4x^2 + 3 \)

We can rewrite this expression as:
\[ x^4 - 4x^2 + 3 = x^4 - 3x^2 - x^2 + 3 \]

Now, we will group the terms:
\[ = (x^4 - 3x^2) - (x^2 - 3) \]

Factoring out \( x^2 \) from the first group:
\[ = x^2(x^2 - 3) - 1(x^2 - 3) \]
\[ = (x^2 - 3)(x^2 - 1) \]

### Step 3: Identify the common factors

Now we have the factored forms:
1. \( x^4 + 3x^2 - 4 = (x^2 + 3)(x^2 - 2) \)
2. \( x^4 - 4x^2 + 3 = (x^2 - 3)(x^2 - 1) \)

The common factor between the two expressions is:
\[ x^2 - 1 \]

### Conclusion

Thus, the HCF of the expressions \( x^4 + 3x^2 - 4 \) and \( x^4 - 4x^2 + 3 \) is:
\[ \text{HCF} = x^2 - 1 \]