Final answer:
Using synthetic division, the polynomial x³ + 5x² − 9x + 3 is evenly divisible by x - 1, with a quotient of x² + 6x - 3 and a remainder of 0.
Step-by-step explanation:
To find the result when x³ + 5x² − 9x + 3 is divided by x - 1 using synthetic division, we perform the following steps:
Write down the coefficients of the polynomial: 1, 5, -9, 3.
Write the zero of the divisor x - 1, which is 1, to the left of a vertical bar.
Bring down the leading coefficient (1) to the bottom row.
Multiply this coefficient by the zero (1) and write the result under the next coefficient (5).
Add the numbers in the second column to get a new number in the bottom row then repeat the multiply and add steps with the new number and the next coefficient.
The final bottom row gives the coefficients of the quotient polynomial and the remainder.
Following these steps gives:
1 | 1 5 -9 3
| 1 6 -3
| ----------------
1 6 -3 0
The quotient is x² + 6x - 3 and the remainder is 0, meaning x³ + 5x² - 9x + 3 is evenly divisible by x - 1.