To solve the division problem [tex]\(\frac{2x^3 - 4x^2 - 3x - 9}{x - 3}\)[/tex], let's use synthetic division. Here is a detailed, step-by-step solution:
1. Identify the coefficients of the polynomial and the root of the divisor:
- The coefficients of the polynomial [tex]\(2x^3 - 4x^2 - 3x - 9\)[/tex] are [tex]\([2, -4, -3, -9]\)[/tex].
- The divisor is [tex]\(x - 3\)[/tex], giving us the root [tex]\(3\)[/tex].
2. Setup for synthetic division:
- We use the root [tex]\(3\)[/tex] and the coefficients [tex]\([2, -4, -3, -9]\)[/tex].
3. Perform the synthetic division:
- Start with the leading coefficient [tex]\(2\)[/tex].
- Multiply the current value by the root and add to the next coefficient.
- Repeat this process until all coefficients have been used.
Let's do this step by step:
- Bring down the leading coefficient:
[tex]\[
2
\][/tex]
- Multiply [tex]\(2\)[/tex] by the root [tex]\(3\)[/tex] and add it to the next coefficient [tex]\(-4\)[/tex]:
[tex]\[
2 \times 3 = 6
\][/tex]
[tex]\[
-4 + 6 = 2
\][/tex]
- Multiply the result [tex]\(2\)[/tex] by the root [tex]\(3\)[/tex] and add it to the next coefficient [tex]\(-3\)[/tex]:
[tex]\[
2 \times 3 = 6
\][/tex]
[tex]\[
-3 + 6 = 3
\][/tex]
- Multiply the result [tex]\(3\)[/tex] by the root [tex]\(3\)[/tex] and add it to the next coefficient [tex]\(-9\)[/tex]:
[tex]\[
3 \times 3 = 9
\][/tex]
[tex]\[
-9 + 9 = 0
\][/tex]
4. Collect the results:
- The synthetic division steps yield:
[tex]\[
[2, 2, 3, 0]
\][/tex]
- The coefficients of the quotient are the first three values (all but the last one):
[tex]\[
[2, 2, 3]
\][/tex]
- The remainder is the last value:
[tex]\[
0
\][/tex]
5. Form the quotient polynomial:
- The quotient polynomial, based on the quotient coefficients [tex]\([2, 2, 3]\)[/tex], is:
[tex]\[
2x^2 + 2x + 3
\][/tex]
- The remainder is [tex]\(0\)[/tex], indicating the division is exact.
6. Conclusion:
- The result of the division is:
[tex]\(\frac{2x^3 - 4x^2 - 3x - 9}{x - 3} = 2x^2 + 2x + 3\)[/tex]
So, the solution to the division problem [tex]\(\frac{2x^3 - 4x^2 - 3x - 9}{x - 3}\)[/tex] is [tex]\(2x^2 + 2x + 3\)[/tex].
