# What is the zeros, degree and end behavior of y=-2x(x-1)(x+5)?

May 30, 2018

Zeroes

$x \setminus \in \setminus \left\{- 5 , 0 , 1 \setminus\right\}$

Degree

Polynomial of third degree

End behaviour

${\lim}_{x \setminus \to + \setminus \infty} - 2 x \left(x - 1\right) \left(x + 5\right) = - \setminus \infty$

${\lim}_{x \setminus \to - \setminus \infty} - 2 x \left(x - 1\right) \left(x + 5\right) = + \setminus \infty$

#### Explanation:

Zeroes

This is very easy: the function is already written in its factorized form. So, if you want to solve

$- 2 x \left(x - 1\right) \left(x + 5\right) = 0$

you are asking for a multiplication to be zero. A multiplication is zero if and only if at least one of its factors is zero, so the alternatives are

• $- 2 x = 0 \setminus \iff x = 0$
• $x - 1 = 0 \setminus \iff x = 1$
• $x + 5 = 0 \setminus \iff x = - 5$

Degree

Just by eyeballing the equation, you can tell this is a polynomial of degree three, since it's the multiplication of three degrees of degree one.

But just to be sure, let's do the actual multiplications:

$\setminus \textcolor{red}{- 2 x \left(x - 1\right)} \left(x + 5\right) = \setminus \textcolor{red}{\left(- 2 {x}^{2} + 2 x\right)} \left(x + 5\right) = - 2 {x}^{3} - 10 {x}^{2} + 2 {x}^{2} + 10 x = - 2 {x}^{3} - 8 {x}^{2} + 10 x$

End Behaviour

The end behaviour is a direct consequence of the degree. If we call any polynomial of even degree ${f}_{e v e n} \left(x\right)$ and any polynomial of odd degree ${f}_{o \mathrm{dd}} \left(x\right)$, the end behaviours will be (assuming the leading term is positive):

${\lim}_{x \setminus \to \setminus \pm \setminus \infty} {f}_{e v e n} \left(x\right) = \setminus \infty$
${\lim}_{x \setminus \to \setminus \pm \setminus \infty} {f}_{o \mathrm{dd}} \left(x\right) = \setminus \pm \setminus \infty$

Since you have a minus sign in front of the polynomial, the limits will be inverted.