<PRE><B>Question 266534</B>
general form of your equation is:


f(x) = ax^3 + bx^2 + cx + d


a is the coefficient of x^3
b is the coefficient of x^2
c is the coefficient of x^1
d is the coefficient of x^0 which makes it the constant term.



the problem states that:


The coefficient of x^2 is 3 less than the coefficient of x^3. 
This means that b = a-3


The coefficient of x is three times the coefficient of x^2. 
This means that c = 3*b


The remaining coefficient is 2 more than the coefficient of x^3. 
This means that d = a+2


The sum of the coefficients is -4.
This means that a + b + c + d = -4.


the equations  you have to work with are:


b = a-3
c = 3*b
d = a+2
a + b + c + d = -4


these wind up being a system of 4 equations in 4 unknowns that have to be solved simultaneously.


this system can be solved by substitution or by elimination.


we will solve by substitution.


start with your final equation of:


a + b + c + d = -4


substitute 3*b for c to get:


a + b + 3*b + d = -4


substitute a-3 for b to get:


a + a - 3 + 3*(a-3) + d = -4


substitute a+2 for d to get:


a + a - 3 + 3*(a-3) + a + 2 = -4


simplify to get:


a + a - 3 + 3*a - 9 + a + 2 = -4


combine like terms to get:


6*a -10 = -4


add 10 to both sides of this equation to get:


6*a = -4 + 10 = 6


divide both sides of this equation to get:


a = 1


your equations that you needed to solve simultaneously are:


b = a-3
c = 3*b
d = a+2
a + b + c + d = -4


since a = 1, you get:


b = 1-3 = -2
c = 3*b = 3*-2 = -6
d = a+2 = 1+2 = 3


your coefficients become:


a = 1
b = -2
c = -6
d = 3


a + b + c + d = -4 becomes:


1 - 2 - 6 + 3 = -4


this becomes:


-8 + 4 = -4 which becomes:


-4 = -4 confirming the values for a,b,c,d are good.


your equation of your polynomial becomes:


f(x) = x^3 -2x^2 -6x + 3


the x^2 coefficient is equal to 3 less than the x^3 coefficient (1-3=-2).
the x coefficient is equal to 3 times the x^2 coefficient (3*(-2)=-6).
the remaining coefficient is equal to 2 more than the x^3 coefficient (1+2=3).


your answer is:


f(x) = x^3 -2x^2 -6x + 3
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