Define a polynomial function k with degree three and has ro

Define a polynomial function, k , with degree three and has roots at x=13 , x=12 , and x=-6 and passes through the point (0,11) .

Define the formula for a parabola (a quadratic function) that has horizontal intercepts (roots) at x=8.3 and x=8.4 and passes through the point (0,5.8) .

Solution

The given roots are x=13,x=12 and x=-6

First step is to write them in factor form

x-13=0,x-12=0 and x+6=0

And to write the polynomial,we have to multiply all the factors together.

k= (x-13)(x-12)(x+6)

First we foil first two factors and on foiling we get

k= (x^2 -25x +156)(x+6)

next is to multiply the terms of the right side of the equation

k= x^3 -19x^2+6x+936

And that\'s the required polynomial

Parabola is of the form

y=ax²+bx+c

In the question the roots are given that is x=8.3 and x=8.4. It means at these two points y=0

First step is to substitute the given values of x and substituting y=0,find out two equations

0= a(8.3)² +b(8.3) +c and 0= a(8.4)²+b(8.4)+c

0= 68.89a + 8.3b+c and   0= 70.56a + 8.4 b + c

And it passes through (0,5.8)

So we plug x=0 and y=5.8 in the above quadratic equation that is y= ax^2 + bx+c

5.8= a(0)²+b(0)+c

5.8= 0 + 0 + c

c=5.8

Next is to substitute the value of c in the above two equations we got

0=68.89a + 8.3 b+5.8 and 0=70.56a + 8.4b +5.8

next step is to move 5.8 to the left side.

-5.8= 68.89a + 8.3b and -5.8=70.56a+ 8.4b

next step is to solve these two equations for a and b

And to solve them for a and b we can use elimination method.

To eliminate b, we multiply equation first by 8.4 and second by -8.3

8.4(-5.8)=8.4(68.89a) + 8.4(8.3)b and -8.3(-5.8)=-8.3(70.56) + (-8.3)(8.4)

-48.72= 578.676 a + 69.72b and 48.14 =-585.648a - 69.72b

Next step is to add them

-48.72 +48.14= 578.676a - 585.648a + 69.72b - 69.72b

-0.58= -6.972a

a=0.083 (approximately)

now we need to find the value of b

And for that we plug the value of a in any of the two equations that have a and b

-5.8=68.89(0.083)+8.3b

-5.8= 5.71787 + 8.3b

-11.51787=8.3b

b=-1.4 approx

so we have a=.083,b=-1.4 and c=5.8

We have to substitute these values in y=ax^2 +bx+c

y=0.083x² - 1.4x + 5.8