Question

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:

Answer 1

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