List all possible rational zeros of...
h(x)= 2x to the (4th power) - 5x (to the third power) + 3x (to the 2nd power) + 4x - 6
Use the Rational Roots Theorem. Any root of the form p/q with p and q relatively prime must be such that p divides 6 and q divides 2. You then find the following possible rational roots:
1/2, 1, 3/2, 2, 3, 6 and
-1/2, -1, -3/2, -2, -3, -6
The possible rational zeros of the polynomial h(x) = 2x^4 - 5x^3 + 3x^2 + 4x - 6 are:
1/2, 1, 3/2, 2, 3, 6
-1/2, -1, -3/2, -2, -3, -6
To find the possible rational zeros of the polynomial h(x), you can use the Rational Roots Theorem. This theorem states that if a polynomial has a rational zero, it must be in the form of p/q, where p divides the constant term (in this case, -6) and q divides the leading coefficient (in this case, 2). The possible rational zeros are the combinations of p and q that satisfy this criterion.
In this case, p can be any divisor of 6 (1, 2, 3, 6), and q can be any divisor of 2 (1, 2). Therefore, you can form all possible rational zeros by dividing each combination of p and q.
For example, when p = 1 and q = 2, the rational zero is 1/2. Similarly, when p = 6 and q = 1, the rational zero is 6.
The possible rational zeros for h(x) are 1/2, 1, 3/2, 2, 3, 6 (positive values), and -1/2, -1, -3/2, -2, -3, -6 (negative values).
Keep in mind that these are only the possible rational zeros. To determine which ones are actually zeros of h(x), you will need to perform polynomial division or use factoring techniques.