Determine the maximum number of real zeros that each polynomial function may have. Then use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros each polynomial function may have. Do not attempt to find the zeros.; f(x)=−4x7+x3−x2+2

Question

Determine the maximum number of real zeros that each polynomial function may have. Then use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros each polynomial function may have. Do not attempt to find the zeros.;  f(x)=−4x7+x3−x2+2

Neunassauk8 · Accepted Answer

Step 1  Given- f(x)=4x7+x3−x2+2  To Find- Determine the maximum number of real zeros that each polynomial function may have. Then use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros each polynomial function may have.   Step 2  Explanation- Rewrite the given expression,  f(x)=4x7+x3−x2+2  As we see that the maximum power on x is , so the maximum number of zeros is 7.  Now, using the Descartes’ Rule , we know that the number of postive roots is equal to the number of sisn changing in the function.  From the given function f(x)=4x7+x3−x2+2, we see that from the sign of + of x3 changes to negative of x2, so there is only one time sign changing in the above function.  So, the number of postive zeros is 1, number of negative zeros is 6.  Answer- Hence, the number of postive zeros is 1, number of negative zeros is 6.

redhotdevil13l3 · Accepted Answer

The maximum number of ral zeros of a polynomial function f is equal to the degree of f.  f(x)=−4x7+x3−x2+2⇒ The maximum number of real zeros is 7 since the degree is 7.  f(x)=−4x7+x3−x2+2⇒  Since there is only 3 changes of signs in the coefficients of f(x), therefore the number of positive zeros can be 3 or 1.  f(−x)=4x7−x3−x2+2⇒  Since there are changes of signs in the coefficients of f(-x), therefore the number of negative zeros can be 2 or 0.

Determine the maximum number of real zeros that each polynomial

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