# Graph each polynomial function. Factor first if the expression is not in factored form. f(x)=x^{2}(x-5)(x+3)(x-1)

Question
Polynomial graphs
Graph each polynomial function. Factor first if the expression is not in factored form. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}$$

2021-02-13
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}$$ The function in factored form The function has four zeros 0,5,-3 and 1. So, the graph of f(x) crossed the x-axis at (0,0), (5,0),(-3,0), and (1,0) To find the y-intercept, substitute 0 for x in f(x) $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}$$
$$\displaystyle{f{{\left({0}\right)}}}={0}^{{{2}}}{\left({0}-{5}\right)}{\left({0}+{3}\right)}{\left({0}-{1}\right)}$$ Substitute 0 for x
$$\displaystyle={0}$$ So, the function f(x) crosses the y-axis at (0,0) Step 2 PSKx^{2}\cdot x \cdot x\cdot x=x^{5} The leading coefficient is 1 Since the leading coefficient is positive and the function f(x) of degree 5 (odd degree) So, the end behavior is $$\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty$$
$$\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow-\infty$$ See the graph below FZP https://q2a.s3-us-west-1.amazonaws.com/dev/1961020641.jpg"/>

### Relevant Questions

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Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Determine whether $$F(x)=5x^{4}-\pi x^{3}+\frac{1}{2}$$ is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term.
$$g(x)=3-\frac{x^{2}}{4}$$
$$G(x)=2(x-3)^{2}(x^{2}+5)$$