Chapter 2.8, Problem 5MRPS

Solution

To find: All zeros of the polynomial function.

The required function would be $f\left(x\right)=x^4+2x^3-5x^2+30$ .

Given information:

A polynomial function: $f\left(x\right)=x^3-4x^2-11x+30$ .

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:

To get the polynomial which has 16 possible rational zeros, find a function whose leading coefficient and constant terms have 16 possible factors in the form $\frac{p}{q}$ .

Take function $f\left(x\right)=x^4+2x^3-5x^2+30$ . The leading coefficient is 1 and the constant term is 30.

Factors of 1 are $\pm 1$ and factors of 30 are $\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 15,\pm 30$ .

  $\begin{array}{c}\frac{p}{q}=\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{3}{1},\pm \frac{5}{1},\pm \frac{6}{1},\pm \frac{10}{1},\pm \frac{15}{1},\pm \frac{30}{1}\\ =\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 15,\pm 30\end{array}$

Upon graphing the function, the required graph would look like as shown below:

  

It can be seen that the graph never crosses the x -axis, so there will be no real solutions for the function.

Therefore, the required function would be $f\left(x\right)=x^4+2x^3-5x^2+30$ .