Chapter EP, Problem 6.8.1EP

To determine

To calculate: The possible rational zeros of the function $f\left(x\right)=3x^5-7x^3-8x+6$ .

Answer to Problem 6.8.1EP

The possible rational zeros of the function $f\left(x\right)=3x^5-7x^3-8x+6$ are $\pm \frac{1}{3},\pm 1,\pm \frac{2}{3},\pm 2,\pm 3,\pm 6$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=3x^5-7x^3-8x+6$ .

Formula used:

Rational zero theorem states that provided a polynomial $P\left(x\right)$ with integral coefficients then every rational zero is of the form $\frac{p}{q}$ that is a rational number in simplest form. Here, p is a factor of the constant term and q is a factor of leading coefficient.

Calculation:

Consider the function $f\left(x\right)=3x^5-7x^3-8x+6$ .

Recall that the Rational zero theorem states that provided a polynomial $P\left(x\right)$ with integral coefficients then every rational zero is of the form $\frac{p}{q}$ that is a rational number in simplest form. Here, p is a factor of the constant term and q is a factor of leading coefficient.

For the provided function leading coefficient is 3 and constant term is 6. Therefore, p is a factor of 6 and q is a factor of 3.

Factors of 3 are $\pm 3\text{ and }\pm 1$ .

Factors of 6 are $\pm 1,\pm 2,\pm 3\text{ and }\pm 6$

The possible combinations of $\frac{p}{q}$ in simplest form are,

  $\frac{p}{q}=\pm \frac{1}{3},\pm 1,\pm \frac{2}{3},\pm 2,\pm 3,\pm 6$

Thus, the possible rational zeros of the function $f\left(x\right)=3x^5-7x^3-8x+6$ are $\pm \frac{1}{3},\pm 1,\pm \frac{2}{3},\pm 2,\pm 3,\pm 6$ .