To find the rational roots of the polynomial f(x) = 2x^3 + x^2 - 7x - 6, we can use the Rational Root Theorem. This theorem states that if a polynomial has a rational root (p/q), where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible root of the polynomial equation.

1. Factors of the constant term (-6): The constant term of the polynomial is -6. To find the factors of -6, we consider all possible combinations of pairs of integers whose product is -6. The factors are: -1, 1, -2, 2, -3, 3, -6, 6.

2. Factors of the leading coefficient (2): The leading coefficient of the polynomial is 2. The factors of 2 are: -1, 1, -2, 2.

3. Possible rational roots: According to the Rational Root Theorem, the possible rational roots of the polynomial f(x) are all the combinations of the factors of the constant term divided by the factors of the leading coefficient. So, our possible rational roots are: ±1/2, ±1, ±2, ±3, ±6.

4. Testing the possible rational roots: We can test each of the possible rational roots using synthetic division or long division to check if they are indeed roots of the polynomial. If the remainder is zero after division, then the tested value is a root.

By testing the possible rational roots, we find that the polynomial f(x) = 2x^3 + x^2 - 7x - 6 has the following rational roots:

-1/2, 1, -2, 3.

Explanation:
- The Rational Root Theorem is a useful tool for finding the rational roots of a polynomial equation.
- By considering the factors of the constant term and the leading coefficient, we can determine the possible rational roots of the polynomial.
- Testing each possible root using synthetic division or long division allows us to verify whether it is a true root of the polynomial.
- In this case, we found that the rational roots of the polynomial f(x) = 2x^3 + x^2 - 7x - 6 are -1/2, 1, -2, and 3.

Let p (x)=( -1)
2 (-1)^3+(-1)^2-7 (-1)-6
-2+1+7-6
0
so x=-1
x+1 is a factor
x+1÷2x^3+x^2-7x-6
we will get
2x^2-x-6
(x+1)(2x^2-x-6)
(x+1)(2x^2-4x+3x-6)
(x+1 ){2x (x-2)+3 (x-2)}
(x+1)(2x+3)(x-2) are factors