 # DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of po avissidep 2021-10-27 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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Step 1
(a). Let a 3rd polynomial function P(x).
If the leading term is positive the end behavior is :
$y\to \mathrm{\infty }$ as $x\to -\mathrm{\infty }$ and y $y\to +\mathrm{\infty }$ as $x\to +\mathrm{\infty }$
If the leading term is negative the end behavior is :
$y\to +\mathrm{\infty }$ as $x\to -\mathrm{\infty }$ and $y\to -\mathrm{\infty }$ as $x\to +\mathrm{\infty }$
As we see in each case the end behavior is different as $x\to -\mathrm{\infty }$ and as $x\to +\mathrm{\infty }$ 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) .
$P\left(x\right)={x}^{4}+{x}^{2}+3$
The terms ${x}^{4}$ and ${x}^{2}$ 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 $P\left(x\right)=\left(x-\sqrt{2}\right)\left(x-\pi \right)\left(x-1\right)$
Step 3
$P\left(x\right)=\left({x}^{2}-x\pi -\sqrt{2}x-\sqrt{2}\pi \right)\left(x-1\right)$
$={x}^{3}-{x}^{2}-\pi {x}^{2}+\pi x-\sqrt{2}{x}^{2}+\sqrt{2}x+\sqrt{2}\pi x-\sqrt{2}\pi$
$={x}^{3}-\left(1+\pi +\sqrt{2}\right){x}^{2}+\left(\pi +\sqrt{2}+\sqrt{2}\pi \right)x-\sqrt{2}\pi$
A 3rd degree polynomial with one rational zero $x=1$ and twp irrational zeros $x=\sqrt{2},\pi$ By our guidelines, we are supposed to answer first three questions only.