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Question 266534: a polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is three times the coefficient of x^2. The remaining coefficient is 2 more than the coefficient of x^3. The sum of the coefficients is -4. Find the polynomial.
Please help. I am so lost on this chapter.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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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