find the equation for the polynomial in the graph 8 5 4 32 2

find the equation for the polynomial in the graph

Solution

From the given graph of K(x), we see that K(x) = 0 for x = 2, x = - 3 and x = - 6.

Therefore (x - 2), (x + 3) and (x + 6) are roots (zeros) of the polynomial K(x).

Thus we can write: K(x) = (x - 2)(x + 3)(x + 6) + c

where c is a constant.

=> K(x) = (x - 2)(x² + 9x + 18) + c

=> K(x) = x³ + 9x² + 18x - 2x² - 18x - 36 + c

=> K(x) = x³ + 7x² - 36 + c

Now at x = 0, K(x) = 6

therefore 6 = 0³  + 7(0)² - 36 + c

=> c = 42

therefore the function is : K(x) =  x³ + 7x² - 36 + 42

=> K(x) =  x³ + 7x² + 6