Chapter 3.4, Problem 45E

To determine

To find: the all rational zeros of polynomial and write the polynomial in factored form.

Answer to Problem 45E

The rational zeros are$2,-1,5,-\frac{1}{3}$

Factored form of polynomial is $P\left(x\right)=\left(x-5\right)\left(x-2\right)\left(x+1\right)\left(3x+1\right)$

Explanation of Solution

Given information:

The polynomial is $P\left(x\right)=3x^5-14x^4-14x^3+36x^2+43x+10$.

Concept used: rational zero theorem,

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

Calculation:

By the rational zeros theorem the rational zeros of $P$are of the form,

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

The constant term is $10$and the leading coefficient is $3$, so

  $\text{possible rational zero of }P=\frac{\text{factor of 10}}{\text{factor of 3}}$

The factors of $10$are $\pm 1,\pm 2,\pm 5,\pm 10$and the factors of $3$ are $\pm 1,\pm 3$.

Thus, the possible rational zeros of of $P$are,$\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{5}{1},\pm \frac{10}{1},\pm \frac{1}{3},\pm \frac{2}{3},\pm \frac{5}{3},\pm \frac{10}{3}$

Simplify the fractions, so the all possible rational zeros:

  $\pm 1,\pm 2,\pm 5,\pm 10,\pm \frac{1}{3},\pm \frac{2}{3},\pm \frac{5}{3},\pm \frac{10}{3}$

For the rational zeros, substitute the x values by the possible rational zeros in the polynomial,

  $\begin{array}{l}P\left(x\right)=3x^5-14x^4-14x^3+36x^2+43x+10\\ P\left(1\right)\ne 0\\ P\left(-1\right)=0\\ P\left(2\right)=0\\ P\left(-2\right)\ne 0\\ P\left(-5\right)\ne 0\\ P\left(5\right)=0\\ P\left(10\right)\ne 0\\ P\left(-10\right)\ne 0\\ P\left(-\frac{1}{3}\right)=0\\ P\left(\frac{1}{3}\right)\ne 0\\ P\left(\frac{2}{3}\right)\ne 0\\ P\left(-\frac{2}{3}\right)\ne 0\\ P\left(\frac{5}{3}\right)\ne 0\\ P\left(-\frac{5}{3}\right)\ne 0\\ P\left(\frac{10}{3}\right)\ne 0\\ P\left(-\frac{10}{3}\right)\ne 0\end{array}$

The rational zeros are $2,-1,5,-\frac{1}{3}$

So the factored polynomial is ,

  $P\left(x\right)=\left(x-5\right)\left(x-2\right)\left(x+1\right)\left(3x+1\right)$