SOLUTION: Find a polynomial of the form f(x)= ax^3+bx^2+cx+d such that f(0)=-5, f(-1)=-6, f(3)=4, and f(4)=-4.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find a polynomial of the form f(x)= ax^3+bx^2+cx+d such that f(0)=-5, f(-1)=-6, f(3)=4, and f(4)=-4.      Log On


   

Question 1206114: Find a polynomial of the form
f(x)= ax^3+bx^2+cx+d
such that f(0)=-5, f(-1)=-6, f(3)=4, and f(4)=-4.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=+ax%5E3%2Bbx%5E2%2Bcx%2Bd
if
f%280%29=-5 => x=0 and f%28x%29=-5
-5=+a%2A0%5E3%2Bb%2A0%5E2%2Bc%2A0%2Bd
d=-5.....eq.1
so far we have f%28x%29=+ax%5E3%2Bbx%5E2%2Bcx%2Bd

f%28-1%29=-6 => x=-1 and f%28x%29=-6
-6=+a%2A%28-1%29%5E3%2Bb%2A%28-1%29%5E2%2Bc%2A%28-1%29-5
-6%2B5=+-a%2Bb-c....solve for+a
a=b-c%2B1.....eq.2

f%283%29=4=> x=3 and f%28x%29=4
4=+a%2A%283%29%5E3%2Bb%2A%283%29%5E2%2Bc%2A%283%29-5
4%2B5=27a%2B9b%2B3c....solve for a
27a=9-9b-3c
a=9%2F27-9b%2F27-3c%2F27
a=1%2F3-b%2F3-c%2F9.....eq.3


f%284%29=-4=> x=4 and f%28x%29=-4
-4=+a%2A%284%29%5E3%2Bb%2A%284%29%5E2%2Bc%2A%284%29-5
-4%2B5=64a++%2B+16b+%2B+4c+
a+=+1%2F64+%28-16b+-+4c+%2B+1%29
a=-b%2F4+-+c%2F16+%2B+1%2F64.......eq.4

from eq.2 and eq.3 we have
b-c%2B1=1%2F3-b%2F3-c%2F9....solve for c
c+=+%283+b%29%2F2+%2B+3%2F4.....eq.5

from eq.2 and eq.4 we have
b-c%2B1=-b%2F4+-+c%2F16+%2B+1%2F64
c+=+%284+b%29%2F3+%2B+21%2F20.....eq.6

from eq.5 and 6 we have
%283+b%29%2F2+%2B+3%2F4=%284+b%29%2F3+%2B+21%2F20...solve for b
b+=+9%2F5

go to eq.5, substitute b
c+=+%283+%289%2F5%29%29%2F2+%2B+3%2F4.....eq.5
c=69%2F20

go to eq.2, substitute b, and c
a=9%2F5-69%2F20%2B1.....eq.2
a=-13%2F20

your function is
f%28x%29=+%28-13%2F20%29x%5E3%2B%289%2F5%29x%5E2%2B%2869%2F20%29x-5