One root of 3x^3-5x^2+2= 0 is 2. Find the remaining roots

To find the remaining roots of the equation 3x^3 - 5x^2 + 2 = 0, we will use synthetic division. Synthetic division is a method used to divide polynomials by a binomial of the form (x - r), where r is a known root of the polynomial. In this case, we will divide by (x - 2) since we know that 2 is one of the solutions.

First, we need to set up the synthetic division table:

2 | 3 -5 0 2

Next, bring down the coefficient of the first term:

2 | 3 -5 0 2
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Then, multiply the divisor (2) by the first term in the dividend (3) and write the product below the next coefficient:

2 | 3 -5 0 2
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6

Next, add the two numbers in the second row:

2 | 3 -5 0 2
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6
-2

Then, multiply the divisor (2) by the newly obtained sum (-2) and write the product below the next coefficient:

2 | 3 -5 0 2
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6 -2
____________
12 -4

Repeat the process of adding and multiplying until all coefficients have been used:

2 | 3 -5 0 2
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6 -2 0 2
____________
12 -4 0

The last number in the bottom row, 0, represents the remainder of the division. The other numbers in the bottom row are the coefficients of the resulting quotient polynomial.

Thus, the quotient polynomial is 3x^2 - 2. To find the remaining roots, we need to solve the equation 3x^2 - 2 = 0.

Rearranging the equation, we have:
3x^2 = 2

Dividing both sides by 3, we get:
x^2 = 2/3

To find the square root of both sides, we have two possible solutions:
x = ±√(2/3)

Therefore, the remaining roots are x = √(2/3) and x = -√(2/3).