Sure, let's go through the process of dividing the polynomial [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex] step-by-step.
Given:
[tex]\[ f(x) = x^4 + 2x^2 + 1 \][/tex]
[tex]\[ d(x) = x + 3 \][/tex]
Our goal is to divide [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex] to find the quotient [tex]\( Q(x) \)[/tex] and the remainder [tex]\( R(x) \)[/tex].
To do this, we use polynomial long division.
1. First Division Step:
- Divide the leading term of [tex]\( f(x) \)[/tex] by the leading term of [tex]\( d(x) \)[/tex]:
[tex]\[ \frac{x^4}{x} = x^3 \][/tex]
- Multiply this result ( [tex]\( x^3 \)[/tex] ) by [tex]\( d(x) \)[/tex]:
[tex]\[ x^3 \cdot (x + 3) = x^4 + 3x^3 \][/tex]
- Subtract this from [tex]\( f(x) \)[/tex]:
[tex]\[ (x^4 + 2x^2 + 1) - (x^4 + 3x^3) = -3x^3 + 2x^2 + 1 \][/tex]
2. Second Division Step:
- Divide the leading term of the new polynomial by the leading term of [tex]\( d(x) \)[/tex]:
[tex]\[ \frac{-3x^3}{x} = -3x^2 \][/tex]
- Multiply this result ( [tex]\( -3x^2 \)[/tex] ) by [tex]\( d(x) \)[/tex]:
[tex]\[ -3x^2 \cdot (x + 3) = -3x^3 - 9x^2 \][/tex]
- Subtract this from the updated polynomial:
[tex]\[ (-3x^3 + 2x^2 + 1) - (-3x^3 - 9x^2) = 11x^2 + 1 \][/tex]
3. Third Division Step:
- Divide the leading term of the new polynomial by the leading term of [tex]\( d(x) \)[/tex]:
[tex]\[ \frac{11x^2}{x} = 11x \][/tex]
- Multiply this result ( [tex]\( 11x \)[/tex] ) by [tex]\( d(x) \)[/tex]:
[tex]\[ 11x \cdot (x + 3) = 11x^2 + 33x \][/tex]
- Subtract this from the updated polynomial:
[tex]\[ (11x^2 + 1) - (11x^2 + 33x) = -33x + 1 \][/tex]
4. Fourth Division Step:
- Divide the leading term of the new polynomial by the leading term of [tex]\( d(x) \)[/tex]:
[tex]\[ \frac{-33x}{x} = -33 \][/tex]
- Multiply this result ( [tex]\( -33 \)[/tex] ) by [tex]\( d(x) \)[/tex]:
[tex]\[ -33 \cdot (x + 3) = -33x - 99 \][/tex]
- Subtract this from the updated polynomial:
[tex]\[ (-33x + 1) - (-33x - 99) = 100 \][/tex]
Now, there are no more [tex]\( x \)[/tex]-terms to divide, so we have:
[tex]\[ Q(x) = x^3 - 3x^2 + 11x - 33 \][/tex]
[tex]\[ R(x) = 100 \][/tex]
Thus, the division can be written as:
[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)}=\frac{x^4+2 x^2+1}{x+3} \\
R(x)=100
\end{array}
\][/tex]
