Question

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?

Polynomial arithmetic
ANSWERED
asked 2021-08-08
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?

Expert Answers (1)

2021-08-09
Step 1
(a). Let a 3rd polynomial function P(x).
If the leading term is positive the end behavior is :
\(\displaystyle{y}\rightarrow\infty\) as \(\displaystyle{x}\rightarrow-\infty\) and y \(\displaystyle{y}\rightarrow+\infty\) as \(\displaystyle{x}\rightarrow+\infty\)
If the leading term is negative the end behavior is :
\(\displaystyle{y}\rightarrow+\infty\) as \(\displaystyle{x}\rightarrow-\infty\) and \(\displaystyle{y}\rightarrow-\infty\) as \(\displaystyle{x}\rightarrow+\infty\)
As we see in each case the end behavior is different as \(\displaystyle{x}\rightarrow-\infty\) and as \(\displaystyle{x}\rightarrow+\infty\) 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) .
\(\displaystyle{P}{\left({x}\right)}={x}^{{{4}}}+{x}^{{{2}}}+{3}\)
The terms \(\displaystyle{x}^{{{4}}}\) and \(\displaystyle{x}^{{{2}}}\) 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 \(\displaystyle{P}{\left({x}\right)}={\left({x}−\sqrt{{{2}}}\right)}{\left({x}-\pi\right)}{\left({x}-{1}\right)}\)
Step 3
\(\displaystyle{P}{\left({x}\right)}={\left({x}^{{{2}}}-{x}\pi-\sqrt{{{2}}}{x}-\sqrt{{{2}}}\pi\right)}{\left({x}-{1}\right)}\)
\(\displaystyle={x}^{{{3}}}-{x}^{{{2}}}-\pi{x}^{{{2}}}+\pi{x}-\sqrt{{{2}}}{x}^{{{2}}}+\sqrt{{{2}}}{x}+\sqrt{{{2}}}\pi{x}-\sqrt{{{2}}}\pi\)
\(\displaystyle={x}^{{{3}}}-{\left({1}+\pi+\sqrt{{{2}}}\right)}{x}^{{{2}}}+{\left(\pi+\sqrt{{{2}}}+\sqrt{{{2}}}\pi\right)}{x}-\sqrt{{{2}}}\pi\)
A 3rd degree polynomial with one rational zero \(\displaystyle{x}={1}\) and twp irrational zeros \(\displaystyle{x}=\sqrt{{{2}}},\pi\) By our guidelines, we are supposed to answer first three questions only.
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