Certainly! Let's start by addressing each part of the problem step-by-step.
### Part (a) Factor [tex]\( f(x) = x^3 - 4x^2 + 64x - 256 \)[/tex] given that [tex]\( 4 \)[/tex] is a zero.
Since [tex]\( 4 \)[/tex] is a known zero of the polynomial [tex]\( f(x) \)[/tex], [tex]\( (x - 4) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
We can use synthetic division to factor [tex]\( f(x) \)[/tex].
Step 1: Set up the synthetic division.
- Write down the coefficients of the polynomial: [tex]\( 1, -4, 64, -256 \)[/tex].
- Use the zero [tex]\( 4 \)[/tex].
Step 2: Perform the synthetic division.
[tex]\[
\begin{array}{r|rrrr}
4 & 1 & -4 & 64 & -256 \\
& & 4 & 0 & 256 \\
\hline
& 1 & 0 & 64 & 0 \\
\end{array}
\][/tex]
Here's how the synthetic division works:
1. Bring down the leading coefficient [tex]\( 1 \)[/tex] to the bottom row.
2. Multiply [tex]\( 4 \)[/tex] (the known zero) by the value just written below the line (1), giving [tex]\( 4 \)[/tex]. Write this result under the next coefficient (-4).
3. Add the column, [tex]\( -4 + 4 = 0 \)[/tex]. Write this result below the line.
4. Repeat the process: multiply [tex]\( 4 \)[/tex] by [tex]\( 0 \)[/tex], getting [tex]\( 0 \)[/tex]. Write this under the next coefficient (64).
5. Add the column, [tex]\( 64 + 0 = 64 \)[/tex]. Write this result below the line.
6. Finally, multiply [tex]\( 4 \)[/tex] by [tex]\( 64 \)[/tex], getting [tex]\( 256 \)[/tex]. Write this under the last coefficient (-256).
7. Add the column, [tex]\( -256 + 256 = 0 \)[/tex]. Write this result below the line.
The bottom row gives the coefficients of the quotient polynomial. Thus, [tex]\( f(x) = (x - 4)(x^2 + 64) \)[/tex].
### Part (b) Solve [tex]\( x^3 - 4x^2 + 64x - 256 = 0 \)[/tex].
Since part (a) tells us that [tex]\( f(x) = (x - 4)(x^2 + 64) \)[/tex], we can now solve the equation by setting each factor to zero.
1. Solve [tex]\( x - 4 = 0 \)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
2. Solve [tex]\( x^2 + 64 = 0 \)[/tex]:
[tex]\[
x^2 + 64 = 0 \implies x^2 = -64 \implies x = \pm \sqrt{-64} \implies x = \pm 8i
\][/tex]
Therefore, the solutions to the equation [tex]\( x^3 - 4x^2 + 64x - 256 = 0 \)[/tex] are:
[tex]\[
x = 4, \quad x = 8i, \quad x = -8i
\][/tex]
In summary:
(a) The factorized form of [tex]\( f(x) \)[/tex] is [tex]\( (x - 4)(x^2 + 64) \)[/tex].
(b) The solutions to the equation [tex]\( x^3 - 4x^2 + 64x - 256 = 0 \)[/tex] are [tex]\( x = 4 \)[/tex], [tex]\( x = 8i \)[/tex], and [tex]\( x = -8i \)[/tex].
