: What is the number of distinct possible rational roots of the polynomial P(x)=5x2+19x−4?
6, 12, 5, or 10
The way the wording is confusing would appreciate the help
P(x)=5x^2+19x−4
is a quadratic. Any quadratic function can have either 0,1, or 2 zeros,
so your choices of answers don't make sense
5x^2 + 19x - 4 = 0
(5x + 1)(x - 4) = 0
x = -1/5 or x = 4
So there are 2 distinct rational roots of the corresponding equations.
I agree that the wording is poor. Equations have roots, functions have zeros.
121
To determine the number of distinct possible rational roots of a polynomial, we can use the Rational Root Theorem. The Rational Root Theorem states that any rational root of a polynomial can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
In the case of the polynomial P(x) = 5x^2 + 19x - 4, the constant term is -4 and the leading coefficient is 5. The factors of -4 are 1, 2, and 4, and the factors of 5 are 1 and 5.
Next, we form all possible combinations of p/q using these factors. These combinations will be the possible rational roots. In this case, these combinations are:
-1/5, -1/4, -1/1, -2/5, -2/4, -2/1, -4/5, -4/4, -4/1
Now, to determine the number of distinct possible rational roots, we count the total number of combinations we obtained. In this case, there are 9 combinations, so the answer is 9.
Therefore, the correct answer is not provided in the options given.