Find a polynomial of degree 3 with real coeffecients and zeros of -3,-1,and 4, for which f(-2)=30

The required polynomial is

SOLUTION:

Given,

Zeroes of a polynomial with degree 3 are – 3, - 1, and 4; f( -2 ) = 30

We have to find the polynomial equation.

Now, we know that, general polynomial equation with degree is

So, let

Now, we know that,

Now, -3 is a root, so

Now, - 1 is a root, so

Now, 4 is a root, so

So, let us solve the above equations to find value of a, b, c, and d

Now, subtract (3) form (1), (2) and (4)

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is