Question

1) What are the real zeroes of x3 + 6 x2 – 9x – 54?

Answer 1

A polynomial with a degree of 3 is understood to have three zeroes or roots. In the given problem in 1, x3 + 6 x2 – 9x – 54, the roots of the polynomial using the calculator are B. B. 3, -3, -6. The same applies for number 2 given the polynomial x3 – 4x2 + 3x – 12. The answer in 2 is D. 4 years. The roots of the polynomial in 3 are: 

x1=−0.45117

x2=1.77474

x3=0.03821+0.70576∗i

x4=0.03821−0.70576∗i. Hence there is A. 1 negative root.

Answer 2

Answer with explanation:

Question 1)

The polynomial function is:

           $f(x)=x^3+6x^2-9x-54$

It could be written as:

$f(x)=x^2(x+6)-9(x+6)\\\\\\i.e.\\\\\\f(x)=(x^2-9)(x+6)$

The roots of a function are the possible values of x where f(x)=0

i.e.

$(x^2-9)(x+6)=0\\\\i.e.\\\\\\(x-3)(x+3)(x+6)=0$

Since,

$(a-b)(a+b)=a^2-b^2$

Hence, we get:

$x+3=0\ or\ x-3=0\ or\ x+6=0\\\\\\i.e.\\\\\\x=-3\ or\ x=3\ or\ x=-6$

Hence, the real zeros of the given problem is:

       B.   3, -3, -6

Question 2)

The profit function is given by:

         $P(x)=x^3-4x^2+3x-12$

We are asked to find the point where the company break even i.e. there is no profit or loss i.e. the possible value of x such that P(x)=0

i.e.

$x^3-4x^2+3x-12=0\\\\\\i.e.\\\\\\x^2(x-4)+3(x-4)\\\\\\i.e.\\\\\\(x^2+3)(x-4)=0$

i.e.

$x^2+3=0\ or\ x-4=0\\\\i.e.\\\\x^2=-3\ or\ x=4$

( As the square of a number can't be negative.

Hence, $x^2=-3$ is not a solution)

Hence, the solution is:

$x=4$

The correct answer is:

      D.  4 years    

Question 3)

The polynomial function is given by:

           $f(x)=5x^4-7x^3-x^2-3x-2$

We know that the number of sign changes of function f(x) gives the possible number of positive real roots

and  the number of sign changes of function f(-x) gives the possible number of negative real roots.

The number of sign changes in f(x) is 1

(Hence, f(x) has at most 1 positive real root)

and $f(-x)=5x^4+7x^3-x^2+3x-2$

The number of sign changes in f(x) is 3

(Hence, f(x) has at most 3 negative real root)

              The correct answer is:

                 C.      3 or 1