Question

the equation below has real roots that are rational. use the rational zero theorem to list all possible rational roots. then graph the polynomial function in the given viewing rectangle to determine which of the possible rational roots are actual roots of the equation.
121x⁴ + 121x³ + 1,330x² - x - 11 = 0; -1,1,0.5 by -14,14,3
list all possible rational roots.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

Explanation:

Step1: Identify the constant term and leading coefficient

The constant term \(p=- 11\), and its factors are \(\pm1,\pm11\). The leading coefficient \(q = 121\), and its factors are \(\pm1,\pm11,\pm121\).

Step2: Apply the Rational Zero Theorem

By the Rational Zero Theorem, the possible rational roots are \(\frac{p}{q}\), so we have \(\pm\frac{1}{1},\pm\frac{1}{11},\pm\frac{1}{121},\pm\frac{11}{1},\pm\frac{11}{11},\pm\frac{11}{121}\).
Simplifying these fractions: \(\pm1,\pm\frac{1}{11},\pm\frac{1}{121},\pm11,\pm1,\pm\frac{1}{11}\).
Removing duplicates, the possible rational roots are \(\pm1,\pm\frac{1}{11},\pm\frac{1}{121},\pm11\).

Answer:

\(-11,-1,-\frac{1}{11},-\frac{1}{121},\frac{1}{121},\frac{1}{11},1,11\)