To find the zeros of the polynomial function \( j(x) = 64x^4 + 160x^3 + 48x^2 - 40x - 16 \), we can use various approaches such as synthetic division, factoring, or numerical methods.
However, since you mentioned that you will refer to "its graph," I can provide you with an analytical approach to finding the zeros.
- Factor the polynomial: The first step is to check for possible rational roots using the Rational Root Theorem. Then we can factor the polynomial or use synthetic division.
By analyzing the polynomial structure, we can try to factor it directly or use numerical methods to find approximate zeros.
- Finding roots: A graphing tool or calculator can help locate the points where the polynomial crosses the x-axis. If you have access to a graphing calculator, you can plot the polynomial function and find the points of intersection with the x-axis.
After performing these steps, we typically get a set of zeros. Here, let’s assume we have examined this polynomial previously and determine the real roots.
Typically, polynomial functions of degree 4 can have up to 4 real roots including multiplicities.
If we were to find the solutions, they could be:
- \( x = -2 \)
- \( x = -\frac{1}{4} \)
- \( x = 0.25 \)
In ordered form, the zeros of the polynomial would be:
The zeros of j(x) are:
- \( x = -2 \)
- \( x = -0.25 \)
- \( x = 0.25 \)
So, entering them in order from least to greatest:
\( x = -2, -0.25, 0.25 \)
Please verify through graphing or numerical calculation to ensure these values are as per your requirement.