Chapter 2, Problem 19TP

Solution

To determine: The number of possible rational zeros of the polynomial function.

There are $16$ possible zeros for the given function

Given information:

A polynomial function: $f\left(x\right)=15x^4+6x^3-8x^2-9$ .

Formula used:

As per Rational Zero Theorem, the possible rational zeros for any function with integer coefficients can be obtained by dividing factors of leading coefficient of highest degree term with factors of the constant term as:

  $\frac{p}{q}=\frac{\text{Factor of constant term }{a}_0}{\text{Factor of leading coefficient term }{a}_n}$

Calculation:

The leading coefficient is $15$ and the constant term is $-9$ .

Factors of $15$ are $\pm 1,\pm 3,\pm 5,\pm 15$ and factors of $-9$ are $\pm 1,\pm 3,\pm 9$ .

  $\frac{p}{q}=\pm \frac{1}{1},\pm \frac{1}{3},\pm \frac{1}{5},\pm \frac{1}{15},\pm \frac{3}{1},\pm \frac{3}{3},\pm \frac{3}{5},\pm \frac{3}{15},\pm \frac{9}{1},\pm \frac{9}{3},\pm \frac{9}{5},\pm \frac{9}{15}$

Now simplify the fractions and remove the duplicates as:

  $\frac{p}{q}=\pm 1,\pm \frac{1}{3},\pm \frac{1}{5},\pm \frac{1}{15},\pm 3,\pm \frac{3}{5},\pm 9,\pm \frac{9}{5}$

Therefore, there are $16$ possible zeros for the given function.