Use the rational zeros theorem to list the potential reational zeros of the polynomial function. Do not attempt to find the zeros.
f(x)=6x^4+9x^3+30x^2+63x+28
To find the potential rational zeros of a polynomial function, we can use the Rational Zeros Theorem.
The Rational Zeros Theorem states that if a polynomial has any rational roots (zeros), they will be in the form of p/q, where p is a factor of the constant term (in this case, 28) and q is a factor of the leading coefficient (in this case, 6).
To determine the factors of 28 and 6, we list all the positive and negative factors of these numbers:
Factors of 28: ±1, ±2, ±4, ±7, ±14, ±28
Factors of 6: ±1, ±2, ±3, ±6
So, the potential rational zeros of the polynomial function f(x) = 6x^4 + 9x^3 + 30x^2 + 63x + 28 could be any of the following:
±1/1, ±2/1, ±4/1, ±7/1, ±14/1, ±28/1, ±1/2, ±2/2, ±4/2, ±7/2, ±14/2, ±28/2, ±1/3, ±2/3, ±4/3, ±7/3, ±14/3, ±28/3, ±1/6, ±2/6, ±4/6, ±7/6, ±14/6, ±28/6.
These are all the potential rational zeros of the given polynomial function. However, it is important to note that not all of these potential zeros will necessarily be actual zeros of the function.
To apply the rational zeros theorem to the polynomial function f(x) = 6x^4 + 9x^3 + 30x^2 + 63x + 28, we need to consider the possible rational zeros.
The rational zeros theorem states that the potential rational zeros are of the form p/q, where p is a factor of the constant term (28), and q is a factor of the leading coefficient (6).
The factors of 28 (the constant term) are ±1, ±2, ±4, ±7, ±14, and ±28.
The factors of 6 (the leading coefficient) are ±1, ±2, ±3, and ±6.
Combining these factors, we get the potential rational zeros:
±1/1, ±2/1, ±4/1, ±7/1, ±14/1, ±28/1,
±1/2, ±2/2, ±4/2, ±7/2, ±14/2, ±28/2,
±1/3, ±2/3, ±4/3, ±7/3, ±14/3, ±28/3,
±1/6, ±2/6, ±4/6, ±7/6, ±14/6, ±28/6.
Therefore, the potential rational zeros of the polynomial function are the numbers listed above.