Answer :
Certainly! Let's perform the polynomial long division of [tex]\( x^2 - 2x + 1 \)[/tex] into [tex]\( x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6 \)[/tex] step by step to find the quotient and remainder.
### Step 1: Setting up the division
We have the dividend [tex]\( x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6 \)[/tex] and the divisor [tex]\( x^2 - 2x + 1 \)[/tex].
### Step 2: Division process
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^5}{x^2} = x^3
\][/tex]
2. Multiply the entire divisor by this quotient term:
[tex]\[
x^3 \cdot (x^2 - 2x + 1) = x^5 - 2x^4 + x^3
\][/tex]
3. Subtract this from the current dividend:
[tex]\[
(x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6) - (x^5 - 2x^4 + x^3) = 2x^4 - 11x^3 + 19x^2 - 14x + 6
\][/tex]
4. Repeat the process: divide [tex]\( 2x^4 \)[/tex] by [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{2x^4}{x^2} = 2x^2
\][/tex]
5. Multiply and subtract:
[tex]\[
2x^2 \cdot (x^2 - 2x + 1) = 2x^4 - 4x^3 + 2x^2
\][/tex]
[tex]\[
(2x^4 - 11x^3 + 19x^2 - 14x + 6) - (2x^4 - 4x^3 + 2x^2) = -7x^3 + 17x^2 - 14x + 6
\][/tex]
6. Continue: divide [tex]\(-7x^3\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-7x^3}{x^2} = -7x
\][/tex]
7. Multiply and subtract again:
[tex]\[
-7x \cdot (x^2 - 2x + 1) = -7x^3 + 14x^2 - 7x
\][/tex]
[tex]\[
(-7x^3 + 17x^2 - 14x + 6) - (-7x^3 + 14x^2 - 7x) = 3x^2 - 7x + 6
\][/tex]
8. Final division step: divide [tex]\(3x^2\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{3x^2}{x^2} = 3
\][/tex]
9. Multiply and subtract once more:
[tex]\[
3 \cdot (x^2 - 2x + 1) = 3x^2 - 6x + 3
\][/tex]
[tex]\[
(3x^2 - 7x + 6) - (3x^2 - 6x + 3) = -x + 3
\][/tex]
### Step 3: Results
- Quotient: [tex]\( x^3 + 2x^2 - 7x + 3 \)[/tex]
- Remainder: [tex]\(-x + 3\)[/tex]
So the quotient is [tex]\( x^3 + 2x^2 - 7x + 3 \)[/tex] and the remainder is [tex]\(-x + 3\)[/tex].
### Step 1: Setting up the division
We have the dividend [tex]\( x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6 \)[/tex] and the divisor [tex]\( x^2 - 2x + 1 \)[/tex].
### Step 2: Division process
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^5}{x^2} = x^3
\][/tex]
2. Multiply the entire divisor by this quotient term:
[tex]\[
x^3 \cdot (x^2 - 2x + 1) = x^5 - 2x^4 + x^3
\][/tex]
3. Subtract this from the current dividend:
[tex]\[
(x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6) - (x^5 - 2x^4 + x^3) = 2x^4 - 11x^3 + 19x^2 - 14x + 6
\][/tex]
4. Repeat the process: divide [tex]\( 2x^4 \)[/tex] by [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{2x^4}{x^2} = 2x^2
\][/tex]
5. Multiply and subtract:
[tex]\[
2x^2 \cdot (x^2 - 2x + 1) = 2x^4 - 4x^3 + 2x^2
\][/tex]
[tex]\[
(2x^4 - 11x^3 + 19x^2 - 14x + 6) - (2x^4 - 4x^3 + 2x^2) = -7x^3 + 17x^2 - 14x + 6
\][/tex]
6. Continue: divide [tex]\(-7x^3\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-7x^3}{x^2} = -7x
\][/tex]
7. Multiply and subtract again:
[tex]\[
-7x \cdot (x^2 - 2x + 1) = -7x^3 + 14x^2 - 7x
\][/tex]
[tex]\[
(-7x^3 + 17x^2 - 14x + 6) - (-7x^3 + 14x^2 - 7x) = 3x^2 - 7x + 6
\][/tex]
8. Final division step: divide [tex]\(3x^2\)[/tex] by [tex]\(x^2\)[/tex]:
[tex]\[
\frac{3x^2}{x^2} = 3
\][/tex]
9. Multiply and subtract once more:
[tex]\[
3 \cdot (x^2 - 2x + 1) = 3x^2 - 6x + 3
\][/tex]
[tex]\[
(3x^2 - 7x + 6) - (3x^2 - 6x + 3) = -x + 3
\][/tex]
### Step 3: Results
- Quotient: [tex]\( x^3 + 2x^2 - 7x + 3 \)[/tex]
- Remainder: [tex]\(-x + 3\)[/tex]
So the quotient is [tex]\( x^3 + 2x^2 - 7x + 3 \)[/tex] and the remainder is [tex]\(-x + 3\)[/tex].
