Answer :
Sure! Let's solve the polynomial division step-by-step.
We are dividing the polynomial [tex]\(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\)[/tex] by [tex]\(x^2 - x + 2\)[/tex].
### Step 1: Divide the Leading Terms
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\( \frac{6x^5}{x^2} = 6x^3 \)[/tex]
2. Multiply the entire divisor by this result:
[tex]\((x^2 - x + 2) \cdot 6x^3 = 6x^5 - 6x^4 + 12x^3\)[/tex]
3. Subtract from the original polynomial:
[tex]\[
(6x^5 + 4x^4 + 11x^3) - (6x^5 - 6x^4 + 12x^3) = 10x^4 - x^3
\][/tex]
### Step 2: Repeat the Process
4. Divide the new leading term:
[tex]\( \frac{10x^4}{x^2} = 10x^2 \)[/tex]
5. Multiply the divisor by this result:
[tex]\((x^2 - x + 2) \cdot 10x^2 = 10x^4 - 10x^3 + 20x^2\)[/tex]
6. Subtract from the new polynomial:
[tex]\[
(10x^4 - x^3 + 23x^2) - (10x^4 - 10x^3 + 20x^2) = 9x^3 + 3x^2
\][/tex]
### Step 3: Continue with Division
7. Divide the next leading term:
[tex]\( \frac{9x^3}{x^2} = 9x \)[/tex]
8. Multiply the divisor by this result:
[tex]\((x^2 - x + 2) \cdot 9x = 9x^3 - 9x^2 + 18x\)[/tex]
9. Subtract from the current polynomial:
[tex]\[
(9x^3 + 3x^2 + 12x) - (9x^3 - 9x^2 + 18x) = 12x^2 - 6x
\][/tex]
### Step 4: Final Steps
10. Divide the next leading term:
[tex]\( \frac{12x^2}{x^2} = 12 \)[/tex]
11. Multiply the divisor by 12:
[tex]\((x^2 - x + 2) \cdot 12 = 12x^2 - 12x + 24\)[/tex]
12. Subtract from the current polynomial:
[tex]\[
(12x^2 - 6x + 16) - (12x^2 - 12x + 24) = 6x - 8
\][/tex]
### Result
The quotient is [tex]\(6x^3 + 10x^2 + 9x + 12\)[/tex] and the remainder is [tex]\(6x - 8\)[/tex].
Thus, the division of the polynomials gives us:
[tex]\[
\left(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\right) \div \left(x^2 - x + 2\right) = 6x^3 + 10x^2 + 9x + 12 \quad \text{remainder: } 6x - 8
\][/tex]
We are dividing the polynomial [tex]\(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\)[/tex] by [tex]\(x^2 - x + 2\)[/tex].
### Step 1: Divide the Leading Terms
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\( \frac{6x^5}{x^2} = 6x^3 \)[/tex]
2. Multiply the entire divisor by this result:
[tex]\((x^2 - x + 2) \cdot 6x^3 = 6x^5 - 6x^4 + 12x^3\)[/tex]
3. Subtract from the original polynomial:
[tex]\[
(6x^5 + 4x^4 + 11x^3) - (6x^5 - 6x^4 + 12x^3) = 10x^4 - x^3
\][/tex]
### Step 2: Repeat the Process
4. Divide the new leading term:
[tex]\( \frac{10x^4}{x^2} = 10x^2 \)[/tex]
5. Multiply the divisor by this result:
[tex]\((x^2 - x + 2) \cdot 10x^2 = 10x^4 - 10x^3 + 20x^2\)[/tex]
6. Subtract from the new polynomial:
[tex]\[
(10x^4 - x^3 + 23x^2) - (10x^4 - 10x^3 + 20x^2) = 9x^3 + 3x^2
\][/tex]
### Step 3: Continue with Division
7. Divide the next leading term:
[tex]\( \frac{9x^3}{x^2} = 9x \)[/tex]
8. Multiply the divisor by this result:
[tex]\((x^2 - x + 2) \cdot 9x = 9x^3 - 9x^2 + 18x\)[/tex]
9. Subtract from the current polynomial:
[tex]\[
(9x^3 + 3x^2 + 12x) - (9x^3 - 9x^2 + 18x) = 12x^2 - 6x
\][/tex]
### Step 4: Final Steps
10. Divide the next leading term:
[tex]\( \frac{12x^2}{x^2} = 12 \)[/tex]
11. Multiply the divisor by 12:
[tex]\((x^2 - x + 2) \cdot 12 = 12x^2 - 12x + 24\)[/tex]
12. Subtract from the current polynomial:
[tex]\[
(12x^2 - 6x + 16) - (12x^2 - 12x + 24) = 6x - 8
\][/tex]
### Result
The quotient is [tex]\(6x^3 + 10x^2 + 9x + 12\)[/tex] and the remainder is [tex]\(6x - 8\)[/tex].
Thus, the division of the polynomials gives us:
[tex]\[
\left(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\right) \div \left(x^2 - x + 2\right) = 6x^3 + 10x^2 + 9x + 12 \quad \text{remainder: } 6x - 8
\][/tex]
