# SOLUTION: 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

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

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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(3504)   (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