SOLUTION: 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 i

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 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 i      Log On


   

Question 206167This question is from textbook Introductory Algebra
: 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.
I have no idea where to start when doing these problems and cannot figure out how to solve these because they are too difficult for me. PLEASE HELP. This was due last night and I have tried and tried, but cannot get the answer. This question is from textbook Introductory Algebra

Found 2 solutions by ankor@dixie-net.com, scott8148:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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%2F6
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

Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
ax^3 + bx^2 + cx + d

"The coefficient of x^2 is 3 less than the coefficient of x^3" ___ b = a - 3

"The coefficient of x is three times the coefficient of x^2" ___ c = 3b = 3a - 9

"The remaining coefficient is 2 more than the coefficient of x^3" ___ d = a + 2

"The sum of the coefficients is -4" ___ a + b + c + d = -4

substituting ___ a + (a - 3) + (3a - 9) + (a + 2) = -4 ___ 6a -10 = -4 ___ a = 1

substituting ___ b = 1 - 3 = -2

substituting ___ c = 3(1) - 9 = -6

substituting ___ d = 1 + 2 = 3

x^3 - 2x^2 - 6x + 3