Question 206167
A polynomial 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.
:
Write coefficients like this:
ax^3 + bx^2 + cx + d
:
Write equation for each statement; we are going to get a, c, d in terms of b:
:
"The coefficient of x^2 is 3 less than the coefficient of x^3."
b = (a-3)
a = (b+3)
:
"The coefficient of x is three times the coefficient of x^2."
c = 3b
:
"The remaining coefficient is 2 more than the coefficient of x^3."
d = (a+2)
Replace a with (b+3)
d = (b+3) + 2
d = (b+5)
:
"The sum of the coefficients is -4."
a + b + c + d = -4
Replace a, c, d using the above equations
(b+3) + b + (3b) + (b+5) = -4
Combine like terms
6b + 8 = -4
6b = -4 - 8
6b = -12
b = {{{-12/6}}}
b -2
:
Use the 1st 3 equations to find a, c, d
a = -2 + 3
a = 1
;
c = 3(-2)
c = -6
:
d = -2 + 5
d = 3
:
 Find the polynomial. x^3 - 2x^2 - 6x + 3