To divide [tex]\(x^4 + 2x^3 + 6x^2 + 4x + 8\)[/tex] by [tex]\(x^2 + 2\)[/tex], we need to perform polynomial long division. Follow these steps:
1. Set Up the Division:
Let's divide [tex]\(x^4 + 2x^3 + 6x^2 + 4x + 8\)[/tex] (the dividend) by [tex]\(x^2 + 2\)[/tex] (the divisor).
2. Divide the Leading Terms:
- Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^2\)[/tex]).
[tex]\[
\frac{x^4}{x^2} = x^2
\][/tex]
This [tex]\(x^2\)[/tex] is the first term of the quotient.
3. Multiply and Subtract:
- Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(x^2 + 2\)[/tex].
[tex]\[
x^2 \cdot (x^2 + 2) = x^4 + 2x^2
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 2x^3 + 6x^2 + 4x + 8) - (x^4 + 2x^2) = 2x^3 + 4x^2 + 4x + 8
\][/tex]
4. Repeat with the New Dividend:
- Now divide [tex]\(2x^3\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{2x^3}{x^2} = 2x
\][/tex]
This [tex]\(2x\)[/tex] is the next term of the quotient.
- Multiply [tex]\(2x\)[/tex] by [tex]\(x^2 + 2\)[/tex]:
[tex]\[
2x \cdot (x^2 + 2) = 2x^3 + 4x
\][/tex]
- Subtract that from the new dividend:
[tex]\[
(2x^3 + 4x^2 + 4x + 8) - (2x^3 + 4x) = 4x^2 + 4x + 8
\][/tex]
5. Repeat Once More:
- Divide [tex]\(4x^2\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{4x^2}{x^2} = 4
\][/tex]
This [tex]\(4\)[/tex] is the final term of the quotient.
- Multiply [tex]\(4\)[/tex] by [tex]\(x^2 + 2\)[/tex]:
[tex]\[
4 \cdot (x^2 + 2) = 4x^2 + 8
\][/tex]
- Subtract that from the new dividend:
[tex]\[
(4x^2 + 4x + 8) - (4x^2 + 8) = 4x + 8 - 8 = 0
\][/tex]
6. Assemble the Quotient and Remainder:
- The quotient is the sum of all the terms we found:
[tex]\[
x^2 + 2x + 4
\][/tex]
- The remainder is the number left over:
[tex]\[
0
\][/tex]
So, the quotient is [tex]\(x^2 + 2x + 4\)[/tex], and the remainder is [tex]\(0\)[/tex].
