Graph the polynomial function. f(x)=−x^{4}+3x^{3}−x+1

What can you say about the graphs of polynomial functions with an even degree compared to the graphs of polynomial functions with an odd degree? Use graphs from the Polynomial Functions Investigation (and maybe some others) to justify your response.

Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $P(x)=a(x-\alpha )(x-\beta )(x-\gamma ),a\ne 0$
Type 2: Two real zeros, one repeated: $P(x)=a(x-\alpha )2(x-\beta ),a\ne 0$
Type 3: One real zero repeated three times: $P(x)=a(x-\alpha )3,a\ne 0$

Type 4: One real and two imaginary zeros: $P(x)=(x-\alpha )(ax2+bx+c),\mathrm{△}=b2-4ac<0,a\ne 0$
Experiment with the graphs of Type 4 cubics. What is the geometrical significance of $\alpha \alpha$ and the quadratic factor which has imaginary zeros?

Graph each polynomial function. Estimate the x-coordinate at which the relative maxima and relative minima occur. State the domain and range for each function. $f(x)=-x^3+4x^2-2x-1$

For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. ${\displaystyle f\left(x\right)=3x-x^3}$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at $x=3$ and $x=1$, and a root of multiplicity 1 at $x=-3$. y-intercept at (0, 9)

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at $x=-5$, $x=-2$, and $x=1$. y-intercept at (0, 6)

Find the cubic polynomial whose graph passes through the points ${\displaystyle (-1,-1),(0,1),(1,3),(4,-1).}$