Graph each polynomial function. f(x)=x^{4}-5x^{2}+6

Graph each polynomial function. f(x)=x^{4}-5x^{2}+6

Question
Graph each polynomial function. \(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{5}{x}^{{{2}}}+{6}\)

Answers (1)

2021-02-12
Step 1
To find the x-intercepts, solve \(\displaystyle{f{{\left({x}\right)}}}={0}\)
\(\displaystyle{x}^{{{4}}}-{5}{x}^{{{2}}}+{6}={0}\)
\(\displaystyle{\left({x}^{{{2}}}-{3}\right)}{\left({x}^{{{2}}}-{2}\right)}={0}\) Factor the trinomial
\(\displaystyle{x}^{{{2}}}-{3}={0}\) or \(\displaystyle{x}^{{{2}}}-{2}={0}\) Set each factor equal to 0
\(\displaystyle{x}=\pm\sqrt{{{3}}}\) or \(\displaystyle{x}=\pm\sqrt{{{2}}}\) Solve each equation
So, the x-intercepts are \(\displaystyle{\left(\sqrt{{{3}}},{0}\right)},{\left(\sqrt{{{2}}},{0}\right)},{\left(-\sqrt{{{3}}},{0}\right)},{\left(-\sqrt{{{2}}},{0}\right)}\)
Step 2
To find the y-intercept evaluate \(\displaystyle{f{{\left({0}\right)}}}\)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{5}{x}^{{{2}}}+{6}\)
\(\displaystyle{f{{\left({0}\right)}}}={0}^{{{4}}}-{5}{\left({0}\right)}^{{{2}}}+{6}\) Substitute 0 for x
\(\displaystyle={6}\) Simplify
So, the y-intercept is \(\displaystyle{\left({0},{6}\right)}\)
Plot the points (-2,2) and (2,2)
image
0

Relevant Questions

asked 2021-03-23
Identify the degree of each term of the polynomial and the degree of the polynomial.
\(\displaystyle-{5}{x}^{{{3}}}+{5}{x}^{{{2}}}+{\frac{{{6}}}{{{7}{x}}}}+{4}\)
asked 2021-06-01
Use the theorems on derivatives to find the derivatives of the following function:
\(\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}\)
asked 2021-05-04
Use the theorems on derivatives to find the derivatives of the following function:
\(\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{{5}}}-{2}{x}^{{{4}}}-{5}{x}+{7}+{4}{x}^{{-{2}}}\)
asked 2021-02-05
Graph each polynomial function. \(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{{3}}}\ -\ {7}{x}^{{{2}}}\ +\ {2}{x}\ +\ {3}\) given that 3 is a zero.
asked 2021-02-10
Express as a polynomial.
\(\displaystyle{\left({3}{x}+{2}\right)}{\left({x}-{5}\right)}{\left({5}{x}+{4}\right)}\)
asked 2021-02-08

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}\ {Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}\) Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial \(\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?

asked 2021-02-21
Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
Find the real zeros of \(f(x) = 5x^{3}\ -\ x^{2}\ +\ 5x\ -\ 1\)
asked 2020-12-25

Polynomial equation with real coefficients that has the roots \(3, 1 -i\ \text{is}\ x^{3} - 5x^{2} + 8x - 6 = 0.\)

asked 2021-03-22
Find all the antiderivatives of the following function. Check your work by taking derivatives.
\(\displaystyle{f{{\left({x}\right)}}}={5}{x}^{{{4}}}\)
asked 2021-06-02
Evaluate the derivatives of the given function for the given value of x.
\(\displaystyle{y}{''}={\left({4}{x}^{{{2}}}-{5}{x}+{4}\right)}{\left({3}-{x}-{2}{x}^{{{2}}}\right)},{x}={5}\)
...