Let's perform the polynomial division step-by-step to find the quotient and the remainder of [tex]\( \frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} \)[/tex].
1. Setup the division:
We want to divide the polynomial [tex]\( 10x^4 - 14x^3 - 10x^2 + 6x - 10 \)[/tex] (the numerator) by [tex]\( x^3 - 3x^2 + x - 2 \)[/tex] (the denominator).
2. First term of the quotient:
[tex]\[
\frac{10x^4}{x^3} = 10x
\][/tex]
Multiply [tex]\( 10x \)[/tex] by the whole denominator:
[tex]\[
10x \cdot (x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x
\][/tex]
Subtract this result from the original numerator:
[tex]\[
(10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x)
\][/tex]
Simplify:
[tex]\[
16x^3 - 20x^2 + 26x - 10
\][/tex]
3. Second term of the quotient:
[tex]\[
\frac{16x^3}{x^3} = 16
\][/tex]
Multiply [tex]\( 16 \)[/tex] by the whole denominator:
[tex]\[
16 \cdot (x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32
\][/tex]
Subtract this result from the remainder:
[tex]\[
(16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32)
\][/tex]
Simplify:
[tex]\[
28x^2 + 10x + 22
\][/tex]
Now, the quotient of the division is [tex]\( 10x + 16 \)[/tex], and the remainder is [tex]\( 28x^2 + 10x + 22 \)[/tex].
Thus, putting it all together:
[tex]\[
\frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} = 10x + 16 + \frac{28x^2 + 10x + 22}{x^3 - 3x^2 + x - 2}
\][/tex]
Finally, filling in the blanks:
- The quotient is [tex]\( \boxed{10} \)[/tex] [tex]\( x + \)[/tex] [tex]\(\boxed{16}\)[/tex]
- The remainder is [tex]\(\boxed{28}\)[/tex] [tex]\(x^2 +\)[/tex] [tex]\(\boxed{10}\)[/tex] [tex]\(x +\)[/tex] [tex]\(\boxed{22}\)[/tex]
