Sure, let's solve the polynomial division of [tex]\((x^2 - 2x + 1)\)[/tex] into [tex]\((x^3 + 4x^2 - 5 + 8)\)[/tex].
First, rewrite the dividend and divisor clearly:
- Dividend (numerator): [tex]\(x^3 + 4x^2 + 3\)[/tex] (since [tex]\(-5 + 8 = 3\)[/tex])
- Divisor (denominator): [tex]\(x^2 - 2x + 1\)[/tex]
We perform polynomial long division step-by-step:
1. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this term and subtract from the dividend:
[tex]\[
(x^2 - 2x + 1) \times x = x^3 - 2x^2 + x
\][/tex]
Subtract this from the original dividend:
[tex]\[
(x^3 + 4x^2 + 3) - (x^3 - 2x^2 + x) = 6x^2 - x + 3
\][/tex]
Now, the new polynomial to divide is [tex]\(6x^2 - x + 3\)[/tex].
3. Divide the new leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{6x^2}{x^2} = 6
\][/tex]
So, the next term of the quotient is [tex]\(6\)[/tex].
4. Multiply the entire divisor by this term and subtract from the new polynomial:
[tex]\[
(x^2 - 2x + 1) \times 6 = 6x^2 - 12x + 6
\][/tex]
Subtract this from the new polynomial:
[tex]\[
(6x^2 - x + 3) - (6x^2 - 12x + 6) = 11x - 3
\][/tex]
Now, the new polynomial to divide is [tex]\(11x - 3\)[/tex].
Since the degree of [tex]\(11x - 3\)[/tex] is less than the degree of the divisor [tex]\((x^2 - 2x + 1)\)[/tex], we stop here and consider [tex]\(11x - 3\)[/tex] the remainder.
Thus, the quotient is [tex]\(x + 6\)[/tex] and the remainder is [tex]\(11x - 3\)[/tex].
Therefore, the result of [tex]\((x^2 - 2x + 1) \div (x^3 + 4x^2 - 5 + 8)\)[/tex] is:
[tex]\[
\text{Quotient: } x + 6
\][/tex]
[tex]\[
\text{Remainder: } 11x - 3
\][/tex]
