Chapter 3.4, Problem 91E

To determine

To find:the rational zeros of the polynomial and plot the graph for the polynomial.

Answer to Problem 91E

The all possible zeros of the polynomial is $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12,\pm \frac{1}{2},\pm 2,\pm \frac{3}{2}$ .

Explanation of Solution

Given information:

the polynomial is $2x^4-5x^3-14x^2+5x+12=0$

concept used:

rational zero theorem,

  $\text{possible rational zero of }P=\frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$

Calculation:

The constant term is 12and the leading coefficient is 2so

  $\text{possible rational zero of }P=\frac{\text{factor of 12}}{\text{factor of 2}}$

The factors of 12are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12$ and the factors of 2 are $\pm 1,\pm 2$ .

Thus, the possible rational zeros of $P$ are,

  $\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{3}{1},\pm \frac{4}{1},\pm \frac{6}{1},\pm \frac{8}{1},\pm \frac{1}{2},\pm \frac{2}{2},\pm \frac{3}{2},\pm \frac{4}{2},\pm \frac{6}{2},\pm \frac{12}{2}$

the possible zeros of $P$ are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12,\pm \frac{1}{2},\pm 2,\pm \frac{3}{2}$ .

The graph obtained as: