Consider the polynomial P(x), shown in standard form and in factored form.



(a) State the behavior at the ends (fill in blanks):
At the left, as x „_ ¡V„V, P(x) „_ __ (choose ¡V„V or „V). At the right, as x „_ „V, P(x) „_ __ (choose ¡V„V or „V).

(b) State the y-intercept:

(c) State the x-intercepts:

(a)

The value of a polynomial at ±&inf; depends only on the sign and order of the highest term.
Polynomials Pn(x) of even order, i.e. n=2,4,.., etc.:
- if the sign of xn is positive, Pn(x)->&inf; as x->±&inf;.
- if the sign of xn is negative, Pn(x)->-&inf; as x->±&inf;.

For example, the shape of a quartic where the coefficient of x4 is positive looks like the letter W, and if the coefficient is negative, it looks like a letter M.

(b) the value of the y-intercept is the value of P(x) when x=0, i.e. equals the constant term.

(c) the value of the x-intercept(s) can be found by factoring the polynomial, and equating each factor to zero.
For example, if P(x)=(x-3)(x-4), we equate (x-3)=0 to get x=3 as one of the two x-intercepts.

To determine the behavior at the ends, the y-intercept, and the x-intercepts of a polynomial given in factored form, we need to identify the factors and their corresponding roots. Let's assume the factored form is P(x) = (x - a)(x - b)(x - c) ... where a, b, c, etc. are the roots of the polynomial.

(a) State the behavior at the ends:
To determine the behavior at the left (as x approaches negative infinity), we consider the degree of the polynomial. If the polynomial has an even degree, the graph will look similar at both ends. If it has an odd degree, the graph will have opposite behaviors at both ends.

So, count the number of factors in the polynomial. If it is an even number, then as x approaches negative infinity (left side), P(x) approaches positive infinity („V). If it is an odd number, P(x) approaches negative infinity (¡V„V) as x approaches negative infinity (left side).

Similarly, to determine the behavior at the right (as x approaches positive infinity), follow the same process. If the degree is even, P(x) approaches positive infinity („V). If the degree is odd, P(x) approaches negative infinity (¡V„V) as x approaches positive infinity (right side).

(b) State the y-intercept:
The y-intercept is the point where the graph of the polynomial intersects the y-axis. To find it, we need to evaluate the polynomial at x = 0. Substitute x = 0 into the polynomial equation, whether in standard or factored form, and solve for P(0). The value obtained will be the y-coordinate of the y-intercept.

(c) State the x-intercepts:
The x-intercepts are the points where the graph of the polynomial intersects the x-axis. To find them, we need to set the polynomial equation equal to zero and solve for x. So, set P(x) = 0 and solve for x. The values obtained will be the x-coordinates of the x-intercepts.

Once you have the polynomial equation in either standard or factored form, you can follow the steps mentioned above to determine the behavior at the ends, the y-intercept, and the x-intercepts.