Chapter 4.2, Problem 60AYU

To determine

To find: The real zeros of $f\left(x\right)=x^4+1.2x^3-7.46x^2-4.692x+15.2881$.

Answer to Problem 60AYU

$\text{The zero is rounded to 0}\text{.99 of 2 decimal places}$.

Explanation of Solution

Given:

$f\left(x\right)=x^4+1.2x^3-7.46x^2-4.692x+15.2881$

$\text{F}$ is a polynomial function.

Step 1: The degree of the polynomial function is 4. There are at most 4 real zeros.

From Descartes’ rule of signs

$f\left(x\right)=x^4+1.2x^3-7.46x^2-4.692x+15.2881$

There will be 2 or 0 positive real zero.

$f\left(-x\right)=x^4+1.2x^3-7.46x^2-4.692x+15.2881$

There will be 2 or 0 negative real zeros.

Step 2: Since all coefficients are non integers, Rational zeros theorem does not apply.

Step 3: Determine the bounds on the zeros of $f$.

Using synthetic division, beginning with $\pm 1$,

$\text{Upper bound = 1}$

$X_{max}=1$

$X\text{-intercepts}$ are one between 4 and 5 and near $-\text{4}$.

The graph between $-\text{2}$ and $-\text{3}$ and between 1 and 2 touches the $x\text{-axis}$.

Therefore the zero is rounded of $0.99$

The zeros is rounded to $0.99$ of 2 decimal places.