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

Answer 1

Zeroes

#x \in \{-5, 0, 1\}#

Degree

Polynomial of third degree

End behaviour

#lim_{x\to+\infty}-2x(x-1)(x+5) = -\infty#

#lim_{x\to-\infty}-2x(x-1)(x+5) = +\infty#

Zeroes

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

#-2x(x-1)(x+5)=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

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:

#\color(red)(-2x(x-1))(x+5) = \color(red)((-2x^2+2x))(x+5) = -2x^3-10x^2+2x^2+10x = -2x^3-8x^2+10x#

End Behaviour

The end behaviour is a direct consequence of the degree. If we call any polynomial of even degree #f_{even}(x)# and any polynomial of odd degree #f_{odd}(x)#, the end behaviours will be (assuming the leading term is positive):
#lim_{x\to\pm\infty}f_{even}(x) = \infty# #lim_{x\to\pm\infty}f_{odd}(x) = \pm\infty#

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

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Answer 2

To find the zeros, degree, and end behavior of the polynomial function (y = -2x(x-1)(x+5)), first, set the function equal to zero and solve for (x):

(-2x(x-1)(x+5) = 0)

The zeros of the function occur when each factor equals zero:

  1. (x = 0) (from (x))
  2. (x = 1) (from (x - 1))
  3. (x = -5) (from (x + 5))

So, the zeros of the function are (x = 0), (x = 1), and (x = -5).

The degree of the polynomial function is determined by the highest power of (x) in the expression, which is 3.

The end behavior of the function can be determined by looking at the leading term of the polynomial. In this case, the leading term is (x^3). As (x) approaches positive infinity ((+\infty)), the leading term (x^3) dominates the function, so the end behavior is as follows:

  • As (x) approaches positive infinity ((+\infty)), (y) approaches positive infinity ((+\infty)).
  • As (x) approaches negative infinity ((-\infty)), (y) approaches negative infinity ((-\infty)).

Therefore, the zeros of the function are (x = 0), (x = 1), and (x = -5), the degree is 3, and the end behavior is as described.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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