Can someone show me how to do this? I got 5 more problems of the same kind. Thannk!
Consider the polynomial P(x), shown in both standard form and factored form.
P(x) = 1/2x^4 - 9/2x^3+21/2x^2+1/2x-15
= 1/2 (x+1)(x-2)(x-3)(x-5)
State the behavior at the ends: (up/down) at the left, and (up/down) at the right?
State the y-intercept
State the x-intercept
Create a graph depicting the polynomial above.
To determine the behavior at the ends, you will need to examine the leading term of the polynomial. In this case, the leading term is 1/2x^4. The degree of the polynomial is even, and the leading coefficient is positive, which means that as x approaches positive or negative infinity, the polynomial will go up (or go in the positive direction) on both ends. Therefore, the behavior at the left end is "up" and the behavior at the right end is also "up".
To find the y-intercept, substitute x = 0 into the polynomial equation. In this case, P(0) = 1/2(0+1)(0-2)(0-3)(0-5) = -15. So the y-intercept is -15.
To find the x-intercepts, set the polynomial equal to 0 and solve for x. In this case, the factored form is already given as 1/2(x+1)(x-2)(x-3)(x-5). Set each factor equal to 0:
x+1 = 0, x-2 = 0, x-3 = 0, x-5 = 0
Solving each equation separately, x = -1, x = 2, x = 3, and x = 5. So the x-intercepts are x = -1, x = 2, x = 3, and x = 5.
To create a graph depicting the polynomial, you will need to plot the y-intercept at -15 and the x-intercepts at -1, 2, 3, and 5. The graph will go up on both ends, so you can draw a smooth curve connecting the intercepts.