Use the Intermediate Value Theorem to show that the polynomial f(x)=x^3+x^2-2x+19 has a real zero between -4 and -1.

Step-by-step explanation:

IVT --> If two points have a known y-value(one positive and one negative) and a known x-value, there has to be at least one real zero in between those two x-values. (assuming f(x) is continuous)

f(-4) = -64+16+8+19 = -21

f(-1) = -1+1+2+19 = 21

Conclusion: Since the -4 and -1 produce a postive y and negative y, there has to be atleast one zero in between.

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2