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To find the other zeros of the polynomial P of x, we will use the fact that the sum of all the zeros of a polynomial is equal to the negative coefficient of the term with the second highest power of x divided by the coefficient of the term with the highest power of x.



Step 1: Write the polynomial in standard form.

P(x) = x^4 - 6x^3 - 26x^2 + 138x - 35

Step 2: Use the sum of zeros formula to find the sum of all the zeros.

sum of zeros = - coefficient of x^3 / coefficient of x^4

sum of zeros = - (-6) / 1

sum of zeros = 6

Step 3: Use the sum of two zeros formula to find the sum of the other two zeros.

sum of two zeros = sum of all zeros - sum of known zeros

sum of two zeros = 6 - (2 + √3 + 2 - √3)

sum of two zeros = 6 - 4

sum of two zeros = 2

Step 4: Use the product of two zeros formula to find the product of the other two zeros.

product of two zeros = constant term / coefficient of x^4

product of two zeros = -35 / 1

product of two zeros = -35

product of known zeros = (2 + √3) (2 - √3) = 1

product of other two zeros = -35 / 1 = -35

Step 5: Use the quadratic formula to find the other two zeros.

x = [-b ± √(b^2 - 4ac)] / 2a

For the polynomial P(x) = x^4 - 6x^3 - 26x^2 + 138x - 35, we can use the quadratic formula for the quadratic equation x^2 + mx - 35 = 0, where m is the sum of the other two zeros.

m = 2

x = [-m ± √(m^2 + 4(35))] / 2

x = [-2 ± √144] / 2

x = [-2 ± 12] / 2

x = -7 or 5

Step 6: Check the answer.

We can check our answer by verifying that the sum and product of all four zeros are equal to 6 and -35, respectively.

sum of all zeros = 2 + √3 + 2 - √3 + (-7) + 5 = 6

product of all zeros = (2 + √3) (2 - √3) (-7) (5) = -35

Therefore, the other two zeros of the polynomial P(x) = x^4 - 6x^3 - 26x^2 + 138x - 35 are -7 and 5.