Question

which of the following choices is not a possible rational root to the polynomial equation?
$f(x)=3x^{4}-11x^{3}+4x^{2}+5x + 7$
$\frac{7}{3}$
3
$-\frac{1}{3}$
-7

Explanation:

Step1: Recall rational root theorem

The rational - root theorem states that if a polynomial equation \(a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0=0\) has a rational root \(p/q\), then \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading - coefficient \(a_n\).
For the polynomial \(f(x)=3x^4−11x^3 + 4x^2+5x + 7\), the leading coefficient \(a_n = 3\) and the constant term \(a_0 = 7\).
The factors of the leading coefficient \(a_n = 3\) are \(\pm1,\pm3\), and the factors of the constant term \(a_0 = 7\) are \(\pm1,\pm7\).

Step2: Determine possible rational roots

The possible rational roots \(p/q\) are of the form \(\pm\frac{1}{1},\pm\frac{1}{3},\pm\frac{7}{1},\pm\frac{7}{3}\).

Answer:

3 is not a possible rational root. So the answer is 3.