Determine all possible rational solutions of the polynomial equation.
-$$f ( x ) = 11 x ^ { 3 } + 19 x ^ { 2 } + 2 x - 22$$
A) $\pm 1 , \pm 2 , \pm 11 , \pm 22$
B) $\pm 1 , \pm \frac { 1 } { 11 } , \pm 2 , \pm 11 , \pm 22$
C) $\pm 1 , \pm \frac { 1 } { 11 } , \pm 2 , \pm \frac { 2 } { 11 } , \pm 11 , \pm 22$
D) $\pm 1 , \pm \frac { 1 } { 2 } , \pm 11 , \pm \frac { 11 } { 2 } , \pm \frac { 1 } { 11 } , \pm \frac { 1 } { 22 }$

Question

Quizplus · Accepted Answer

To find all possible rational solutions of the given polynomial equation, we use the Rational Root Theorem. This theorem states that if a polynomial has rational roots, then each root, when written in lowest terms, is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. For the given polynomial $f(x) = 11x^3 + 19x^2 + 2x - 22$, the constant term is -22 and the leading coefficient is 11. The factors of -22 are ±1, ±2, ±11, ±22, and the factors of 11 are ±1, ±11. Therefore, the possible rational roots are the factors of -22 divided by the factors of 11, which gives: ±1, ±2, ±11, ±22, and when divided by 11, we get ±1/11, ±2/11. Thus, the complete list of possible rational solutions is ±1, ±1/11, ±2, ±2/11, ±11, ±22.

Determine All Possible Rational Solutions of the Polynomial Equation f(x)=11x3+19x2+2x−22f ( x ) = 11 x ^ { 3 } + 19 x ^ { 2 } + 2 x - 22f(x)=11x3+19x2+2x−22

Determine All Possible Rational Solutions of the Polynomial Equation $f ( x ) = 11 x ^ { 3 } + 19 x ^ { 2 } + 2 x - 22$