Chapter 4.2, Problem 31AYU

In Problems 33-44, determine the maximum number of real zeros that each polynomial function may have. Then list the potential rational zeros of each polynomial function. Do not attempt to find the zeros.

43. $f\left(x\right)\text{ = 6}{x}^4\text{ + 2}{x}^3\text{ - }{x}^2\text{ + 20}$

To determine

Maximum Number of real positive or negative real zeros of the given polynomial function using Descartes’ Rule of signs and the potential rational zeros of each polynomial function using rational zeros theorem.

Answer to Problem 31AYU

The degree of the polynomial is 4. Therefore the number of real zeros by real zero theorem can be at most $=4$.

Therefore there will be 2 positive real zero or no real positive real zeros.

Therefore there can be 2 negative real zeros or no real negative real zeros.

The rational zeros $=\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 10,\pm 20,\pm \frac{1}{2},\pm \frac{1}{6}\text{, }\pm \frac{2}{3},\pm \frac{1}{3},\pm \frac{4}{3},\pm \frac{5}{2},\pm \frac{5}{3},\pm \frac{5}{6},\pm \frac{10}{3},\pm \frac{20}{3}$.

Explanation of Solution

Given:

$f\left(x\right)=6x^4+2x^3-x^2+20$

The degree of the polynomial is 4. Therefore the number of real zeros by real zero theorem can be at most $=4$.

$f\left(x\right)=6x^4+2x^3-x^2+20$

$+$ to $-,-$ to $+$

There are two variations of signs of nonzero coefficient of $f\left(x\right)$ by Descartes’ Rule of signs;

Number of positive real zeros $=$ number of variations of signs of $f\left(x\right)$ or number of variations less than even integer.

Therefore there will be 2 positive real zero or no real positive real zeros.

Let us consider $f\left(-x\right)$ by replacing $x$ by $-x$.

$f\left(-x\right)=6{\left(-x\right)}^4+2{(-x)}^3-{(-x)}^2+20$

$f\left(-x\right)=6x^4-2x^3-x^2+20$

$+$ to $-,-$ to $+$

There are two variations of signs of nonzero coefficient of $f\left(-x\right)$ by Descartes’ Rule of signs;

Number of negative real zeros $=$ number of variations of signs or number of variations less than even integer.

Therefore there can be two negative real zeros or no real negative zeros.

Rational zeros theorem provides information about the rational zeros of a polynomial function with integer coefficients.

If $\frac{p}{q}$ in its lowest terms is a rational zero of $f$, then $p$ is a factor of $a_0$ and $q$ is the factor of $a_n$.

$f\left(x\right)=6x^4+2x^3-x^2+20$

Here $a_0=\text{ 20}$ and $a_n=6$.

Zeros of $6$, $q=\pm 1,\pm 2,\pm 3,\pm 6$.

Zeros of 20, $p=\pm 1,\pm 2,\pm 4,\pm 5,\pm 10,\pm 20$.

The potential rational zeros $=\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6,\pm 10,\pm 20,\pm \frac{1}{2},\pm \frac{1}{6}\text{, }\pm \frac{2}{3},\pm \frac{1}{3},\pm \frac{4}{3},\pm \frac{5}{2},\pm \frac{5}{3},\pm \frac{5}{6},\pm \frac{10}{3},\pm \frac{20}{3}$.