is -1 a real coefficient

Question

So I'm being asked to write a polynomial function with real coefficients with zeros -1, 2 ,and -3. The function given is F(x)=xcubed+ blankxsquared+x-6. I think the answer is 2 but not so sure. Help me?

 

Answer 1

If a polynomial function F(x) has zeros at points a, b and c then this means that (x-a), (x-b) and (x-c) are all factors of F(x), which means that F(x) can be written as some multiple of (x-a)*(x-b)*(x-c).  In this case F(x) is a multiple of 

 

(x+1)*(x-2)*(x+3) = 

(x+1)*(x^2 + x - 6) = 

x^3 +x^2 - 6x + x^2 + x - 6 = 

x^3 +2x^2 - 5x - 6 

 

So you are correct, the coefficient of x^2 is 2, but it seems that the linear term should be -5x, not +x.  You can check the answer by evaluating F at the points -1, 2 and -3, which should all be 0.  

 

F(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0

F(2) = (2)^3 + 2(2)^2 - 5(2) - 6 = 8 + 8 - 10 - 6 = 0

F(-3) = (-3)^3 + 2(-3)^2 - 5(-3) - 6 = -27 + 18 + 15 - 6 = 0