SOLUTION: The graph of the polynomial function f(x)- Ax^3+Bx^2+Cx+D, where and A,B,C,and D are integers, has 3 x-intercepts at 1,2,and -6. If the graph passes thought the point M(3,18),

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The graph of the polynomial function f(x)- Ax^3+Bx^2+Cx+D, where and A,B,C,and D are integers, has 3 x-intercepts at 1,2,and -6. If the graph passes thought the point M(3,18),       Log On


   

Question 102977: The graph of the polynomial function f(x)- Ax^3+Bx^2+Cx+D,
where and A,B,C,and D are integers, has 3 x-intercepts at 1,2,and -6.
If the graph passes thought the point M(3,18),
determine the values of A,B,C and D.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
An x intercept is a where f(x)=0.
1.f%281%29=0
2.f%282%29=0
3.f%28-6%29=0
and you know that
4.f%283%29=18
Using the equations you get
1.f%281%29=0
A%2A1%5E3%2BB%2A1%5E2%2BC%2A1%2BD=0
A%2BB%2BC%2BD=0
2.f%283%29=0
A%2A2%5E3%2BB%2A2%5E2%2BC%2A2%2BD=0
8A%2B4B%2B2C%2BD=0
3.f%28-6%29=0
A%2A%28-6%29%5E3%2BB%2A%28-6%29%5E2%2BC%2A%28-6%29%2BD=0
-216%2AA%2B36%2AB-6%2AC%2BD=0
4.f%283%29=18
A%2A%283%29%5E3%2BB%2A%283%29%5E2%2BC%2A%283%29%2BD=18
27%2AA%2B9%2AB%2B3%2AC%2BD=18
Four equations, four unknowns.
Since you didn't specify a method to choose, I'll choose a matrix method, if that's OK.
These equations can be represented as three matrices G, U, and V.

U=%28%0D%0A+matrix%28+4%2C1%2C+%0D%0A++A%2CB%2CC%2CD%0D%0A+%29%0D%0A%29
V=%28%0D%0A+matrix%28+4%2C1%2C+%0D%0A++0%2C0%2C0%2C18%0D%0A+%29%0D%0A%29
where [G]x[U]=[V]
Then [U]=[G]inv x [V]
Find the inverse of [G](using EXCEL) and multiply by [V]
The inverse of G is

Since [V] has all values of zero except the last value. The solution matrix,[U], is 18 multplied by the last column of the inverse matrix or
A=1
B=3
C=-16
D=12