To divide the polynomial [tex]\( f(x) \)[/tex] by the polynomial [tex]\( d(x) \)[/tex], we perform polynomial long division. Given:
[tex]\[ f(x) = 3x^3 - 5x^2 + x + 1 \][/tex]
[tex]\[ d(x) = -x + 4 \][/tex]
We want to express the division [tex]\( \frac{f(x)}{d(x)} \)[/tex] in the form of [tex]\( \frac{f(x)}{d(x)} = Q(x) + \frac{R(x)}{d(x)} \)[/tex], where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R(x) \)[/tex] is the remainder.
Let's perform polynomial long division step-by-step:
### Step 1: Division
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{3x^3}{-x} = -3x^2 \][/tex]
2. Multiply the entire divisor [tex]\( d(x) \)[/tex] by this quotient term [tex]\(-3x^2\)[/tex]:
[tex]\[ (-3x^2)(-x + 4) = 3x^3 - 12x^2 \][/tex]
3. Subtract this result from [tex]\( f(x) \)[/tex]:
[tex]\[ 3x^3 - 5x^2 + x + 1 - (3x^3 - 12x^2) = 3x^3 - 5x^2 + x + 1 - 3x^3 + 12x^2 = 7x^2 + x + 1 \][/tex]
### Step 2: Continue the process
1. Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[ \frac{7x^2}{-x} = -7x \][/tex]
2. Multiply the entire divisor [tex]\( d(x) \)[/tex] by [tex]\(-7x\)[/tex]:
[tex]\[ (-7x)(-x + 4) = 7x^2 - 28x \][/tex]
3. Subtract this result from the polynomial obtained after the first step:
[tex]\[ 7x^2 + x + 1 - (7x^2 - 28x) = 7x^2 + x + 1 - 7x^2 + 28x = 29x + 1 \][/tex]
### Step 3: Continue the process
1. Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[ \frac{29x}{-x} = -29 \][/tex]
2. Multiply the entire divisor [tex]\( d(x) \)[/tex] by [tex]\(-29\)[/tex]:
[tex]\[ (-29)(-x + 4) = 29x - 116 \][/tex]
3. Subtract this result from the polynomial obtained after the second step:
[tex]\[ 29x + 1 - (29x - 116) = 29x + 1 - 29x + 116 = 117 \][/tex]
### Conclusion of the division process
The quotient [tex]\( Q(x) \)[/tex] is:
[tex]\[ Q(x) = -3x^2 - 7x - 29 \][/tex]
The remainder [tex]\( R(x) \)[/tex] is:
[tex]\[ R(x) = 117 \][/tex]
Putting it all together:
[tex]\[ \frac{3x^3 - 5x^2 + x + 1}{-x + 4} = -3x^2 - 7x - 29 + \frac{117}{-x + 4} \][/tex]
So, the remainder [tex]\( R(x) \)[/tex] is:
[tex]\[ R(x) = 117 \][/tex]
Thus, the final answer is:
[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)}=\frac{3 x^3-5 x^2+x+1}{-x+4} \\
R(x) = 117
\end{array}
\][/tex]
