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<h2>Descartes Rule of Sign</h2><h3>Answer:</h3>
There are Real Roots for the Equation
According to Descartes Rule of Sign, The maximum number of real roots a polynomial equation has is the number of times the sign of term changes from a higher power term to a lower power term.
In the given equation, , we can already see that from
to
we can see that the sign didn't change so there might be no real roots. From
to
, we can see that the signs change so we now know that there might be
real root for equation. From
to
, the signs changed so there might be
real roots. From
to
, it changed again so there might be
real roots. There are no other term with lesser power to compare for
so we are left with
real roots.
Answer:
The polynomial of minimum degree is .
Step-by-step explanation:
According to the statement, we appreciate that polynomial pass through the following points:
(i) , (ii)
, (iii)
, (iv)
From Algebra we know that any n-th grade polynomial can be constructed by knowing n+1 different points. Hence, the polynomial of minimum degree is a quartic function. The polynomial has the following form:
We proceed to expand the expression until standard form is obtained:
If we know that , then:
Then, the polynomial is:
The polynomial of minimum degree is .