DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

snowlovelydayM 2021-08-08 Answered
DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial.
(a) A polynomial of degree 3 that has no real zeros
(b) A polynomial of degree 4 that has no real zeros
(c) A polynomial of degree 3 that has three real zeros, only one of which is rational
(d) A polynomial of degree 4 that has four real zeros, none of which is rational
What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?
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Expert Answer

Tasneem Almond
Answered 2021-08-09 Author has 91 answers
Step 1
(a). Let a 3rd polynomial function P(x).
If the leading term is positive the end behavior is :
y as x and y y+ as x+
If the leading term is negative the end behavior is :
y+ as x and y as x+
As we see in each case the end behavior is different as x and as x+ which means that the graph of a third degree polynomial must cross the x-axis at least once.
This means that every polynomial P(x) of degree 3 has at least one real zero.
Step 2
(b). Let a 4th degree polynomial P(x) .
P(x)=x4+x2+3
The terms x4 and x2 are non negative as even powers of x and the constant term is positive.
P(x) is strictly positive. So the polynomial P(x) has no real zeros.
(c). Let the 3rd degree polynomial P(x)=(x2)(xπ)(x1)
Step 3
P(x)=(x2xπ2x2π)(x1)
=x3x2πx2+πx2x2+2x+2πx2π
=x3(1+π+2)x2+(π+2+2π)x2π
A 3rd degree polynomial with one rational zero x=1 and twp irrational zeros x=2,π By our guidelines, we are supposed to answer first three questions only.
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