Answer :
To find the zeros of the function [tex]\( f(x) = 3x^4 - 13x^3 - 56x^2 + 164x - 48 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
1. Identify the Polynomial:
The function given is a fourth-degree polynomial:
[tex]\[
f(x) = 3x^4 - 13x^3 - 56x^2 + 164x - 48
\][/tex]
2. Finding Zeros:
The zeros of [tex]\( f(x) \)[/tex] are the solutions to the equation [tex]\( f(x) = 0 \)[/tex]. Using algebraic techniques, such as synthetic division, factoring, or numerical methods, we find that the zeros of this polynomial are:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = \frac{1}{3} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 6 \)[/tex]
3. Verification:
You can verify these solutions by substituting each zero back into the polynomial and checking that it equates to zero, thus confirming these values are indeed the correct zeros of [tex]\( f(x) \)[/tex].
By following these steps, we find the zeros of the function are [tex]\( x = -4, \frac{1}{3}, 2, \)[/tex] and [tex]\( 6 \)[/tex].
1. Identify the Polynomial:
The function given is a fourth-degree polynomial:
[tex]\[
f(x) = 3x^4 - 13x^3 - 56x^2 + 164x - 48
\][/tex]
2. Finding Zeros:
The zeros of [tex]\( f(x) \)[/tex] are the solutions to the equation [tex]\( f(x) = 0 \)[/tex]. Using algebraic techniques, such as synthetic division, factoring, or numerical methods, we find that the zeros of this polynomial are:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = \frac{1}{3} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = 6 \)[/tex]
3. Verification:
You can verify these solutions by substituting each zero back into the polynomial and checking that it equates to zero, thus confirming these values are indeed the correct zeros of [tex]\( f(x) \)[/tex].
By following these steps, we find the zeros of the function are [tex]\( x = -4, \frac{1}{3}, 2, \)[/tex] and [tex]\( 6 \)[/tex].
