To divide the polynomial [tex]\(10x^4 - 14x^3 - 10x^2 + 6x - 10\)[/tex] by the polynomial [tex]\(x^3 - 3x^2 + x - 2\)[/tex], we can follow the process of polynomial long division. Here is the step-by-step solution:
1. Setup the division:
[tex]\[
\frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2}
\][/tex]
2. First term of the quotient:
[tex]\[
\frac{10x^4}{x^3} = 10x
\][/tex]
Multiply [tex]\(10x\)[/tex] by the divisor [tex]\(x^3 - 3x^2 + x - 2\)[/tex]:
[tex]\[
10x(x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x
\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x)
\][/tex]
[tex]\[
10x^4 - 14x^3 - 10x^2 + 6x - 10 - 10x^4 + 30x^3 - 10x^2 + 20x
\][/tex]
[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 divisor [tex]\(x^3 - 3x^2 + x - 2\)[/tex]:
[tex]\[
16(x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32
\][/tex]
Subtract this result from the remaining polynomial:
[tex]\[
(16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32)
\][/tex]
[tex]\[
16x^3 - 20x^2 + 26x - 10 - 16x^3 + 48x^2 - 16x + 32
\][/tex]
[tex]\[
28x^2 + 10x + 22
\][/tex]
So the final result of our division is:
- Quotient: [tex]\(10x + 16\)[/tex]
- Remainder: [tex]\(28x^2 + 10x + 22\)[/tex]
Thus, the quotient is [tex]\(10x + 16\)[/tex] and the remainder is [tex]\(28x^2 + 10x + 22\)[/tex].
[tex]\[
\frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} = 10x + 16 \quad \text{with a remainder of} \quad 28x^2 + 10x + 22
\][/tex]
