Sketch, find the end behavior(EB), y- int, zeros, and multiplicities?

$y = \frac{1}{36} {\left({x}^{2} - 6\right)}^{2} {\left(x + 1\right)}^{2} \left({x}^{2} - 3 x - 4\right) {\left(3 - 2 x\right)}^{3} \left({x}^{2} + 1\right) \left({x}^{2} - 4 x - 1\right)$

Dec 17, 2017

The highest degree is $14$, and the leading coefficient is positive, therefore, the ${\lim}_{x \to - \infty} = {\lim}_{x \to + \infty} = + \infty$.

The y-intercept can be found by plugging in $x = 0$, getting $y = 108$.

The zeroes can be found by setting the function equal to $0$ and solving for $x$. I won't do all of them but for instance if

$0 = \frac{1}{36} {\left({x}^{2} - 6\right)}^{2} \left(x - 2\right)$

Then ${x}^{2} - 6 = 0 \to x = \pm \sqrt{6}$ and $x = 2$.

Multiplicities are the degree of each $0$. For instance, in $y = \frac{1}{36} {\left({x}^{2} - 6\right)}^{2} \left(x - 2\right)$ , then ${\left({x}^{2} - 6\right)}^{2}$ has degree $2$ and $x - 2$ has degree $1$, because of the exponents.

Hopefully this helps!