Answer :
To find all zeros of the polynomial function [tex]\( f(x) = 12x^4 - 71x^3 + 70x^2 + 128x - 160 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
Here's a breakdown of how this can be done:
1. Rational Root Theorem: This theorem proposes that any rational root, expressed in the form [tex]\( \frac{p}{q} \)[/tex], must have [tex]\( p \)[/tex] (a factor of the constant term, [tex]\(-160\)[/tex]) and [tex]\( q \)[/tex] (a factor of the leading coefficient, [tex]\(12\)[/tex]). We would list out all possible values of [tex]\( \frac{p}{q} \)[/tex] using these factors and test them.
2. Synthetic Division or Polynomial Division: Once we test for potential roots from our list and find a root, we use synthetic division or polynomial division to divide the polynomial by [tex]\( (x - \text{found root}) \)[/tex]. This helps reduce the degree of the polynomial and find further roots.
3. Continue until Quadratic or Linear Polynomial: Repeat the process until the polynomial is reduced to a quadratic or linear polynomial, where solving roots is straightforward, using methods like factoring, the quadratic formula, or simple solving for linear terms.
For this function, after performing this method, we find that the zeros (roots) are:
- [tex]\(-\frac{4}{3}\)[/tex]
- [tex]\(\frac{5}{4}\)[/tex]
- [tex]\(2\)[/tex]
- [tex]\(4\)[/tex]
These values are the points at which the polynomial equals zero, making them the solutions or zeros of the function [tex]\( f(x) \)[/tex].
Here's a breakdown of how this can be done:
1. Rational Root Theorem: This theorem proposes that any rational root, expressed in the form [tex]\( \frac{p}{q} \)[/tex], must have [tex]\( p \)[/tex] (a factor of the constant term, [tex]\(-160\)[/tex]) and [tex]\( q \)[/tex] (a factor of the leading coefficient, [tex]\(12\)[/tex]). We would list out all possible values of [tex]\( \frac{p}{q} \)[/tex] using these factors and test them.
2. Synthetic Division or Polynomial Division: Once we test for potential roots from our list and find a root, we use synthetic division or polynomial division to divide the polynomial by [tex]\( (x - \text{found root}) \)[/tex]. This helps reduce the degree of the polynomial and find further roots.
3. Continue until Quadratic or Linear Polynomial: Repeat the process until the polynomial is reduced to a quadratic or linear polynomial, where solving roots is straightforward, using methods like factoring, the quadratic formula, or simple solving for linear terms.
For this function, after performing this method, we find that the zeros (roots) are:
- [tex]\(-\frac{4}{3}\)[/tex]
- [tex]\(\frac{5}{4}\)[/tex]
- [tex]\(2\)[/tex]
- [tex]\(4\)[/tex]
These values are the points at which the polynomial equals zero, making them the solutions or zeros of the function [tex]\( f(x) \)[/tex].
