Please check my answers. Thanks!

Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.

5.2x^3-13x^2+22x-8=0
-My answer:{1/2,2,4}

6.x^3-8x^2-x+8=0
-My answer: {1,-1,8}

Find the rational zero of the polynomial function and use it to find all the zeros of the function.

3. f(x)=x^3+3x^2-4x-12
My answer: {-3,-2,2}

4. f(x)3x^3-14x^2+13x+6
My answer: {-1/3,2,3}

you are correct in all cases

To check your answers for polynomial equations and find the roots, we can use synthetic division and the rational root theorem.

1. For the equation 5.2x^3 - 13x^2 + 22x - 8 = 0:
To test the possible rational roots, we factorize the constant term and the leading coefficient and find their factors:
Possible factors of constant term (8): ±1, ±2, ±4, ±8
Possible factors of leading coefficient (5.2): ±1, ±2, ±5, ±10, ±13, ±26, ±65, ±130

Testing each combination of factors using synthetic division, we find the following results:
1/2: Not a root
2: Not a root
4: Yes, it is a root

Therefore, the correct answer is {4}.

2. For the equation x^3 - 8x^2 - x + 8 = 0:
Using the same method, we test the possible rational roots:
Possible factors of constant term (8): ±1, ±2, ±4, ±8
Possible factors of leading coefficient (1): ±1

Testing each combination of factors using synthetic division:
1: Not a root
-1: Yes, it is a root
8: Not a root

Therefore, the correct answer is {-1}.

3. For the polynomial function f(x) = x^3 + 3x^2 - 4x - 12:
The rational root theorem tells us that the possible rational roots are the factors of the constant term (12) divided by the factors of the leading coefficient (1). Testing the values for f(x) = 0, we find:
-3: Yes, it is a root
-2: Yes, it is a root
2: Yes, it is a root

Therefore, the correct answer is {-3, -2, 2}.

4. For the polynomial function f(x) = 3x^3 - 14x^2 + 13x + 6:
Using the rational root theorem, we find the possible rational roots:
Possible factors of constant term (6): ±1, ±2, ±3, ±6
Possible factors of leading coefficient (3): ±1, ±3

Testing each combination of factors:
-1/3: Yes, it is a root
2: Yes, it is a root
3: Yes, it is a root

Therefore, the correct answer is {-1/3, 2, 3}.

It is important to note that these answers should always be checked by substituting them back into the original equation to verify if they indeed satisfy the given equation.