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
Arrange the polynomial in descending order
Write down all the factors of the constant term. These are all the possible values of p .
Write down all the factors of the leading coefficient. These are all the possible values of q .
Write down all the possible values of . Remember that since factors can be negative, and - must both be included. Simplify each value and cross out any duplicates.
Use synthetic division to determine the values of for which P() = 0 . These are all the rational roots of P(x) .
factors of 28: 28,1; 14,7;and the negatives of those
facors of 6: 6,1;3,2 and the negatives of those.
now list all the p/q values, eliminating the duplicates.
To use the Rational Zeros Theorem to list the potential rational zeros of the polynomial function, we need to consider the factors of the constant term divided by the factors of the leading coefficient.
The Rational Zeros 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 potential rational zero.
In our case, the polynomial function is f(x) = 6x^4 + 9x^3 + 30x^2 + 63x + 28.
The constant term of the polynomial is 28, and the leading coefficient is 6.
Factors of the constant term (28) are: ±1, ±2, ±4, ±7, ±14, ±28.
Factors of the leading coefficient (6) are: ±1, ±2, ±3, ±6.
So, the potential rational zeros of the polynomial function f(x) = 6x^4 + 9x^3 + 30x^2 + 63x + 28 are the fractions that can be formed by taking a factor of the constant term divided by a factor of the leading coefficient.
These potential rational zeros are:
±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
Please note that these are potential rational zeros, and we would need to test them further using other techniques such as synthetic division or graphing methods to determine the actual zeros of the polynomial function.