Expand `(a-4)^(3)`
A. `a^(3)-12a^(2) +48a +64`
B. `a^(3) -48a^(2) +12 a -64`
C. `a^(3) +12 a^(2) -48a -64`
D. `a^(3) -12 a^(2) + 48a -64`

Options

A. C
B. A
C. D
D. B

Answer

To expand \((a - 4)^3\), we can use the binomial expansion formula for the cube of a binomial, which is given by:

\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]

In our case, \(a\) is \(a\) and \(b\) is \(4\). Therefore, we can substitute \(a\) and \(b\) into the formula:

1. **Calculate \(a^3\)**:
\[ a^3 = a^3 \]

2. **Calculate \(-3a^2b\)**:
\[ -3a^2b = -3a^2(4) = -12a^2 \]

3. **Calculate \(3ab^2\)**:
\[ 3ab^2 = 3(a)(4^2) = 3(a)(16) = 48a \]

4. **Calculate \(-b^3\)**:
\[ -b^3 = -4^3 = -64 \]

Now, we can combine all these terms together:

\[ (a - 4)^3 = a^3 - 12a^2 + 48a - 64 \]

Thus, the expanded form of \((a - 4)^3\) is:

\[ a^3 - 12a^2 + 48a - 64 \]

Now, let's match this with the given options:

A. \(a^3 - 12a^2 + 48a + 64\)
B. \(a^3 - 48a^2 + 12a - 64\)
C. \(a^3 + 12a^2 - 48a - 64\)
D. \(a^3 - 12a^2 + 48a - 64\)

The correct answer is **D**: \(a^3 - 12a^2 + 48a - 64\).