The root of x³ - 3x + 2 is x = 1.
Further factoring reveals that the complete factorization is (x - 1)²(x + 2).
Here's how to find the root using synthetic division:
Set up the synthetic division tableau:
Write the coefficients of the polynomial in descending order: 1, 0, -3, 2.
Place a potential root (we'll start with 1) outside the tableau to the left.
Bring down the first coefficient:
Bring down the 1 to the first row of the quotient.
Multiply and add:
Multiply the root (1) by the first number in the quotient (1) and write the result (1) below the second coefficient (0).
Add the numbers in the second column (0 + 1) and write the result (1) in the third row.
Repeat for remaining coefficients:
Multiply 1 by 1 and write the result (1) below -3.
Add -3 + 1 to get -2.
Multiply 1 by -2 and write the result (-2) below 2.
Add 2 - 2 to get 0.
Interpret the resultt
The quotient is 1x² + 1x - 2.
The remainder is 0.
Since the remainder is 0, 1 is a root of the polynomial.
Therefore, the root of x³ - 3x + 2 is x = 1.
Further factoring reveals that the complete factorization is (x - 1)²(x + 2).