Let's solve the problem of dividing the polynomial [tex]\(-x^4 + 2x^3 + 6x^2 - 10x - 4\)[/tex] by [tex]\(x - 2\)[/tex] step-by-step.
We begin with polynomial long division.
1. Division:
- The leading term of the divisor [tex]\(x - 2\)[/tex] is [tex]\(x\)[/tex].
- The leading term of the dividend [tex]\(-x^4 + 2x^3 + 6x^2 - 10x - 4\)[/tex] is [tex]\(-x^4\)[/tex].
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{-x^4}{x} = -x^3\)[/tex].
2. Multiplication and Subtraction:
- Multiply the entire divisor [tex]\(x - 2\)[/tex] by [tex]\(-x^3\)[/tex]: [tex]\((-x^3)(x - 2) = -x^4 + 2x^3\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(-x^4 + 2x^3 + 6x^2 - 10x - 4) - (-x^4 + 2x^3) = 6x^2 - 10x - 4.
\][/tex]
3. Repeat the Process:
- Now consider the new polynomial [tex]\(6x^2 - 10x - 4\)[/tex].
- The leading term of [tex]\(6x^2 - 10x - 4\)[/tex] is [tex]\(6x^2\)[/tex].
- Divide by the leading term of the divisor [tex]\(x\)[/tex]: [tex]\(\frac{6x^2}{x} = 6x\)[/tex].
4. Multiplication and Subtraction:
- Multiply the entire divisor [tex]\(x - 2\)[/tex] by [tex]\(6x\)[/tex]: [tex]\((6x)(x - 2) = 6x^2 - 12x\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(6x^2 - 10x - 4) - (6x^2 - 12x) = -10x + 12x - 4 = 2x - 4.
\][/tex]
5. Repeat the Process Again:
- Now consider the new polynomial [tex]\(2x - 4\)[/tex].
- The leading term of [tex]\(2x - 4\)[/tex] is [tex]\(2x\)[/tex].
- Divide by the leading term of the divisor [tex]\(x\)[/tex]: [tex]\(\frac{2x}{x} = 2\)[/tex].
6. Multiplication and Subtraction:
- Multiply the entire divisor [tex]\(x - 2\)[/tex] by [tex]\(2\)[/tex]: [tex]\((2)(x - 2) = 2x - 4\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(2x - 4) - (2x - 4) = 0.
\][/tex]
Thus, the division results in the quotient:
[tex]\[
-x^3 + 6x + 2
\][/tex]
and the remainder is:
[tex]\[
0
\][/tex]
So, [tex]\(-x^4 + 2x^3 + 6x^2 - 10x - 4\)[/tex] divided by [tex]\(x - 2\)[/tex] gives the quotient [tex]\(-x^3 + 6x + 2\)[/tex].
Thus, the correct answer is:
B. [tex]\(-x^3 + 6x + 2\)[/tex]
