To find [tex]\( f(x) \div g(x) \)[/tex] where [tex]\( f(x) = x^4 + 64 \)[/tex] and [tex]\( g(x) = x^2 - 4x + 8 \)[/tex], we need to perform polynomial division. Let's go through the steps involved in this process.
### Step 1: Setup the Division
We begin by setting up the polynomial division:
[tex]\[ \frac{x^4 + 64}{x^2 - 4x + 8} \][/tex]
### Step 2: Perform the Division
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{x^4}{x^2} = x^2 \][/tex]
So, the first term in the quotient is [tex]\( x^2 \)[/tex].
2. Multiply [tex]\( x^2 \)[/tex] by [tex]\( g(x) \)[/tex] and subtract from [tex]\( f(x) \)[/tex]:
[tex]\[ x^2 \cdot (x^2 - 4x + 8) = x^4 - 4x^3 + 8x^2 \][/tex]
Now, perform the subtraction:
[tex]\[ (x^4 + 0x^3 + 0x^2 + 0x + 64) - (x^4 - 4x^3 + 8x^2) = 4x^3 - 8x^2 + 64 \][/tex]
3. Repeat the process with the new polynomial [tex]\( 4x^3 - 8x^2 + 64 \)[/tex]:
Divide the leading term:
[tex]\[ \frac{4x^3}{x^2} = 4x \][/tex]
So, the next term in the quotient is [tex]\( 4x \)[/tex].
4. Multiply [tex]\( 4x \)[/tex] by [tex]\( g(x) \)[/tex] and subtract:
[tex]\[ 4x \cdot (x^2 - 4x + 8) = 4x^3 - 16x^2 + 32x \][/tex]
Perform the subtraction:
[tex]\[ (4x^3 - 8x^2 + 0x + 64) - (4x^3 - 16x^2 + 32x) = 8x^2 - 32x + 64 \][/tex]
5. Repeat the process with the new polynomial [tex]\( 8x^2 - 32x + 64 \)[/tex]:
Divide the leading term:
[tex]\[ \frac{8x^2}{x^2} = 8 \][/tex]
So, the next term in the quotient is [tex]\( 8 \)[/tex].
6. Multiply [tex]\( 8 \)[/tex] by [tex]\( g(x) \)[/tex] and subtract:
[tex]\[ 8 \cdot (x^2 - 4x + 8) = 8x^2 - 32x + 64 \][/tex]
Perform the subtraction:
[tex]\[ (8x^2 - 32x + 64) - (8x^2 - 32x + 64) = 0 \][/tex]
### Step 3: Conclusion
After performing the division, the quotient is [tex]\( x^2 + 4x + 8 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
Thus, the result of [tex]\( \frac{f(x)}{g(x)} \)[/tex] is:
[tex]\[ \boxed{x^2 + 4x + 8} \][/tex]
