Final answer:
The correct options are:
- the y int = 0
- the graph has same end behavior as y = -3x
- there are 3 real Zeros
- the degree is five
Step-by-step explanation:
The given polynomial function is f(x) = -2x(x + 1)²(x - 3)(x² + 4).
Let's analyze the statements:
1) The y-intercept is 0: To find the y-intercept, let x = 0. Therefore, y = -2(0)(0 + 1)²(0 - 3)(0² + 4) = 0. So, the statement is true.
2) The graph bounces off the x-axis: The graph bounces off the x-axis when there is an odd multiplicity zero at a particular x-value.
Looking at the function, we see that the multiplicity of each factor is even, so the graph does not bounce off the x-axis. So, the statement is false.
3) The graph has the same end behavior as y = -3x: The end behavior is determined by the leading term, which is -2x⁵.
Therefore, the end behavior is the same as y = -3x. So, the statement is true.
4) There are 2 imaginary zeros: To find the zeros, we set f(x) = 0. Solving the equation, we find that there are no complex solutions. So, the statement is false.
5) There are 3 real zeros: To find the zeros, we set f(x) = 0. By factoring the function, we can see that there are indeed 3 real zeros: x = 0, x = -1, and x = 3. So, the statement is true.
6) The degree is five: The degree of a polynomial is determined by the highest exponent in the polynomial. In this case, the highest exponent is 5, so the degree is indeed five. So, the statement is true.