Question

What does the graph of y = (x + 2)(x + 1)(x – 3)2 do near the point (3, 0)?
The graph is( above or below) the x-axis to its left, then( passes through the point or is tangent to the x-axis at the point),and is(above or below) the x-axis to its right.

Answer 1

The real (that is, the non-complex) zeroes of a polynomial correspond to the x-intercepts of the graph of that polynomial. So we can find information about the number of real zeroes of a polynomial by looking at the graph and, conversely, we can tell how many times the graph is going to touch or cross the x-axis by looking at the zeroes of the polynomial (or at the factored form of the polynomial).

A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. For instance, the quadratic (x + 3)(x – 2) has the zeroes x = –3 and x = 2, each occuring once. The eleventh-degree polynomial (x + 3)4(x – 2)7 has the same zeroes as did the quadratic, but in this case, the x = –3 solution has multiplicity 4 because the factor (x + 3) occurs four times (that is, the factor is raised to the fourth power) and the x = 2 solution has multiplicity 7 because the factor (x – 2) occurs seven times.

Answer 2

Answer:

The graph is above the x-axis to (3,0) left and is tangent to the x-axis at the point and above the x-axis to its right.

Step-by-step explanation:

The equation of the polynomial is

$y=(x+2)(x+1)(x-3)^2$

The degree of function is 4.

Put y=0, to find the zeros of the function.

$0=(x+2)(x+1)(x-3)^2$

$x=-2,-1,3$

The zeros of the function are -2, -1 and -3 (with multiplicity 2).

If a zero has even multiplicity, then the function is tangent to the x-axis at that point. It means the function will bounce back.

for x>3,

f(x)>0

So, the graph is above that x-axis to the left or rigth (3,0).

Therefore the he graph is above the x-axis to (3,0) left and is tangent to the x-axis at the point and above the x-axis to its right.