Factorise: `y ^(16) - 63 y ^(8) - 64`

Options

A. `(y ^(8) - 1) (y ^(4) + 8) ( y ^(4) - 8)`
B. `(y ^(4) + 8) ^(2) ( y ^(8) +1)`
C. `(y ^(4) - 8) ^(2) (y ^(8) -1)`
D. `( y ^(4) + 8) (y ^(4) - 8) (y ^(8) +1)`

Answer

To factorize the expression \( y^{16} - 63y^{8} - 64 \), we can follow these steps:

### Step 1: Substitute \( t = y^8 \)
Let’s set \( t = y^8 \). Then, the expression becomes:
\[ t^2 - 63t - 64 \]

### Step 2: Factor the quadratic expression
We need to factor \( t^2 - 63t - 64 \). We are looking for two numbers that multiply to \(-64\) (the constant term) and add to \(-63\) (the coefficient of \( t \)). The numbers that satisfy this condition are \(-64\) and \(1\).

Thus, we can write:
\[ t^2 - 63t - 64 = (t - 64)(t + 1) \]

### Step 3: Substitute back \( t = y^8 \)
Now we substitute back \( t = y^8 \):
\[ (y^8 - 64)(y^8 + 1) \]

### Step 4: Factor \( y^8 - 64 \)
Notice that \( y^8 - 64 \) is a difference of squares:
\[ y^8 - 64 = (y^4)^2 - (8)^2 = (y^4 - 8)(y^4 + 8) \]

### Step 5: Final expression
Now we can write the complete factorization:
\[ (y^4 - 8)(y^4 + 8)(y^8 + 1) \]

### Final Answer
Thus, the factorization of \( y^{16} - 63y^{8} - 64 \) is:
\[ (y^4 - 8)(y^4 + 8)(y^8 + 1) \]

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