Factorise: `a ^(4) - 20 a ^(2) + 64`

Options

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

Answer

To factorise the expression \( a^4 - 20a^2 + 64 \), we can follow these steps:

### Step 1: Recognize the form of the expression
The expression \( a^4 - 20a^2 + 64 \) is a quadratic in terms of \( a^2 \). We can let \( x = a^2 \). Thus, the expression can be rewritten as:
\[ x^2 - 20x + 64 \]

### Step 2: Factor the quadratic expression
Next, we need to factor the quadratic \( x^2 - 20x + 64 \). We look for two numbers that multiply to \( 64 \) (the constant term) and add up to \( -20 \) (the coefficient of \( x \)). The numbers that satisfy this are \( -16 \) and \( -4 \).

So, we can write:
\[ x^2 - 20x + 64 = (x - 16)(x - 4) \]

### Step 3: Substitute back for \( x \)
Now, we substitute back \( x = a^2 \):
\[ (a^2 - 16)(a^2 - 4) \]

### Step 4: Factor further using the difference of squares
Both factors can be further factored using the difference of squares:
1. \( a^2 - 16 = (a - 4)(a + 4) \)
2. \( a^2 - 4 = (a - 2)(a + 2) \)

Thus, we have:
\[ (a - 4)(a + 4)(a - 2)(a + 2) \]

### Final Answer
The fully factored form of the expression \( a^4 - 20a^2 + 64 \) is:
\[ (a - 4)(a + 4)(a - 2)(a + 2) \]

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