Question

Without graphing the function, describe the graph of p(x) = x(x – 6)2(x + 5)2. Your description should include degree, end behavior, zeros (including multiplicity), and number of turns.

Answer 1

We have a polynomial p(x):

`p(x)=x(x-6)^2(x+5)^2`

This is a fifth degree polynomial as the greatest exponent of x, when we distribute all the factors, will be 5.

It will have a zero at x = 0, x = 6 (double) and x = -5 (double).

When x tends to minus infinity, as the degree of p(x) is odd and the coefficient for x^5 will be positive, then p(x) will also tend to minus infinity.

In the same way, we can conclude that when x approaches infinity, p(x) will also approach infinity.

The number of turns can be found by looking at the roots and extreme values.

We start from very negative values of x.

Then, we will have the first turn (turn = 1) when x = -5.

As the root at x = -5 is double, p(x) will decrease for values x > -5 and then increase again because we have another root at x = 0. Then, we have another turn (turns = 2) between x =-5 and x = 0.

As x = 0 is not a multiple root, we can assume that p(x) increases until a maximum point and then decreases to the double root x = 6. So we have another turn (turns = 3) between x = 0 and x = 6.

As x = 6 is a double root, p(x) will increase for x > 6, so we have the last turn after x =6 (turns = 4).

Answer:

Degree = 5

Behaviour: when x approaches minus infinity, p(x) approaches minus infinity.

When x approaches infinity, p(x) approaches infinity.

Zeros: x = -5 (double), x = 0, x = 6 (double)

Number of turns: 4