Let's divide the polynomials [tex]\(a(x) = 5x^3 + 2x^2 + x + 2\)[/tex] by [tex]\(b(x) = x^3 + x + 1\)[/tex] to find the quotient polynomial [tex]\(q(x)\)[/tex] and the remainder polynomial [tex]\(r(x)\)[/tex].
The general form for polynomial division is:
[tex]\[
\frac{a(x)}{b(x)} = q(x) + \frac{r(x)}{b(x)}
\][/tex]
where the degree of the remainder polynomial [tex]\(r(x)\)[/tex] is less than the degree of the divisor [tex]\(b(x)\)[/tex].
1. We start with the highest degree terms of both polynomials. The highest degree term of [tex]\(a(x)\)[/tex] is [tex]\(5x^3\)[/tex], and the highest degree term of [tex]\(b(x)\)[/tex] is [tex]\(x^3\)[/tex].
2. Divide the leading term of [tex]\(a(x)\)[/tex] by the leading term of [tex]\(b(x)\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the first term of the quotient [tex]\(q(x)\)[/tex] is [tex]\(5\)[/tex].
3. Now, multiply [tex]\(5\)[/tex] by [tex]\(b(x)\)[/tex]:
[tex]\[
5 \times (x^3 + x + 1) = 5x^3 + 5x + 5
\][/tex]
4. Subtract this result from [tex]\(a(x)\)[/tex]:
[tex]\[
(5x^3 + 2x^2 + x + 2) - (5x^3 + 5x + 5) = 2x^2 + x - 5x + 2 - 5 = 2x^2 - 4x - 3
\][/tex]
The resulting polynomial [tex]\(2x^2 - 4x - 3\)[/tex] has a degree less than [tex]\(b(x)\)[/tex], so it will be the remainder [tex]\(r(x)\)[/tex].
Therefore, the quotient polynomial [tex]\(q(x)\)[/tex] is:
[tex]\[
q(x) = 5
\][/tex]
And the remainder polynomial [tex]\(r(x)\)[/tex] is:
[tex]\[
r(x) = 2x^2 - 4x - 3
\][/tex]
So the final results are:
[tex]\[
q(x) = 5
\][/tex]
[tex]\[
r(x) = 2x^2 - 4x - 3
\][/tex]
