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.

(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.