Question

1.) Solve the equation12x^3+x^2-9/2 x-1=)given that -1/2 is a root

2.) Use decartes Rule of signs to determine the possible number of positive and negative real zero's for, f(x)=-3x^4+2x^3-6x^2+x+4

3.) The equation for f is given by the simplified expression that results after performing he indicated operation. Write the equation for f and then graph the function
1-2/x+1
__________
1+1/x-1

4.) The force that it requires to strech a spring varies directly with the distance that is is streched. If a pound force can stretch a spring 8 inches, how much force would it take to stret

Answer 1

I'll help you through 1.
Given that -1/2 is a root, use synthetic division to find (12x^3+x^2-9/2 x-1)/(x+1/2)

Answer 2

2) Descartes' rule of signs state that number of +ve real roots of a polynomial arranged in descending power of exponents is equal to sign changes of its coefficient.
Here, f(x)=-3x^4+2x^3-6x^2+x+4
No. of sign changes = 3
so, no. of +ve real roots is 3.

Now, as degree of this polynomial is 4, it can have a max. of 4 real roots, if 3 are +ve then the remaining one must be -ve.

Answer 3

4) Thats called Hooke's law. It has an equation similar to this form : F = -k.x
[ Force is proportional to distance to which spring is streched ]

You can substitute 1st set of given information [ A pound can stretch 8 inches as F and x ] get the value of k and get other required values.