Problem 77 - College Algebra

Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros of \(f(x)-x^{5}-x^{4}+x^{3}-x^{2}+x-8 .\) Verify your result by using a graphing utility to graph \(f\)

Step-by-Step Solution

Answer

The possible number of positive real zeros is 1, 3, or 5 and the possible number of negative real zeros is 0 or 2. This can be verified using a graphing utility.

Step 1: Determine possible number of positive real zeros

Replace \(x\) in \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\) by \(+x\) and count the number of sign changes to get the possible number of positive real zeros. The sign sequence is (+,-,+,-,+,-), therefore there are 5 sign changes. So, there are 5 or 5-2=3 or 3-2=1 positive real zeros.

Step 2: Determine possible number of negative real zeros

Now replace \(x\) in \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\) by \(-x\), to get the possible number of negative real zeros. This gives \(-x^5 - x^4 - x^3 + x^2 - x + 8\). The sign sequence is (-,-,-,+,-,+), resulting in 2 sign changes. So, there could be 2 or 2-2=0 negative real roots.

Step 3: Verify with a Graphing Utility

You can confirm your results using a graphing utility. Plot the function \(f(x)=x^{5}-x^{4}+x^{3}-x^{2}+x-8\), and observe the number of times the graph intersects the x-axis. This represents the number of real roots. Compare this with your calculated results for verification.