Answer :
To find the real zeros of the polynomial function [tex]\( f(x) = x^8 + 10x^7 - 20x^6 - 60x^5 + 70x^4 + 60x^3 - 20x^2 - 10x + 1 \)[/tex], you can follow these steps:
1. Graph the Function: Use a graphing calculator to plot the polynomial function [tex]\( f(x) \)[/tex]. This will give you a visual representation of where the function crosses the x-axis.
2. Identify Approximate Zeros: Observe the points where the graph intersects the x-axis. These points are the approximate real zeros of the function.
3. Use the Root-Finding Feature: Most graphing calculators have a root-finding feature (also called 'zero' or 'solve'). Use this feature to find the precise zeros. Generally, you need to select an interval around each suspected zero to get a more precise value.
4. Refine the Zeros: The calculator will provide numerical results for the zeros. Ensure these results are accurate and round them to three decimal places.
Following these steps, the real zeros of the function [tex]\( f(x) = x^8 + 10x^7 - 20x^6 - 60x^5 + 70x^4 + 60x^3 - 20x^2 - 10x + 1 \)[/tex] are:
[tex]\[
-11.258, -2.125, -0.647, -0.426, 0.089, 0.470, 1.546, 2.350
\][/tex]
Thus, the real zeros of the function, rounded to three decimal places, are:
[tex]\[
-11.258, -2.125, -0.647, -0.426, 0.089, 0.470, 1.546, 2.350
\][/tex]
1. Graph the Function: Use a graphing calculator to plot the polynomial function [tex]\( f(x) \)[/tex]. This will give you a visual representation of where the function crosses the x-axis.
2. Identify Approximate Zeros: Observe the points where the graph intersects the x-axis. These points are the approximate real zeros of the function.
3. Use the Root-Finding Feature: Most graphing calculators have a root-finding feature (also called 'zero' or 'solve'). Use this feature to find the precise zeros. Generally, you need to select an interval around each suspected zero to get a more precise value.
4. Refine the Zeros: The calculator will provide numerical results for the zeros. Ensure these results are accurate and round them to three decimal places.
Following these steps, the real zeros of the function [tex]\( f(x) = x^8 + 10x^7 - 20x^6 - 60x^5 + 70x^4 + 60x^3 - 20x^2 - 10x + 1 \)[/tex] are:
[tex]\[
-11.258, -2.125, -0.647, -0.426, 0.089, 0.470, 1.546, 2.350
\][/tex]
Thus, the real zeros of the function, rounded to three decimal places, are:
[tex]\[
-11.258, -2.125, -0.647, -0.426, 0.089, 0.470, 1.546, 2.350
\][/tex]
