If f(x)=6x^3 - ax^2 + bx + c when a, b and c are constant when f(x) is divided by x^2 - 4 the remainder is 23x - 26. When f(x) divided by (x +3) the remainder is -220. Find:
(A) the value of a, b and c
(B) the solution of the equation f
(x)=0
A little division shows that
6x^3-ax^2+bx+c / (x+4)
has a remainder of -16a-4b+c-384
6x^3-ax^2+bx+c/(x^2-4)
has a remainder of (b+24)x+c-4a
So,
-16a-4b+c-384 = -220
b+24 = 23
c-4a = -26
solve those and you get
f(x) = 6x^3 + 31/2 x^2 - x - 88
You can verify the remainders for yourself.
thank you steve! But how to find the value of a and c?
To find the value of a, b, and c, we need to use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by another polynomial (x - k), the remainder is equal to f(k).
(A) Finding the value of a, b, and c:
1. The given polynomial is f(x) = 6x^3 - ax^2 + bx + c.
2. When f(x) is divided by x^2 - 4, the remainder is 23x - 26.
So, we have: f(x) = (x^2 - 4)(x + k) + (23x - 26).
Expand the right side of the equation to get: f(x) = x^3 + kx^2 - 4x - 4k + 23x - 26.
Simplify it further: f(x) = x^3 + kx^2 + 23x - 4x - 4k - 26.
Comparing the coefficients, we can see that:
Coefficient of x^3 is 1, so the coefficient of x^2 in (x + k) must also be 1.
Coefficient of x^2 is -a, so -a = k.
Coefficient of x is 23, so 23 = -4 + k.
Coefficient of constant term is -4k - 26, so -4k - 26 = -26.
Solve these equations to find the value of k and subsequently the values of a, b, and c.
From the second equation, k = -a.
Substitute this value of k in the third equation: 23 = -4 - a.
Rearranging the equation, we have a = -27.
Now substitute the value of a in the fourth equation: -4k - 26 = -26.
Simplifying it further: -4k = 0.
Therefore, k = 0, and consequently, a = -27.
3. When f(x) is divided by (x + 3), the remainder is -220.
So, we have: f(x) = (x + 3)(x + k) + (-220).
Expand the right side of the equation to get: f(x) = x^2 + kx + 3x + 3k - 220.
Simplify it further: f(x) = x^2 + (k + 3)x + 3k - 220.
Comparing the coefficients, we can see that:
Coefficient of x^2 is 6, so the coefficient of x in (x + k) must also be 6.
Coefficient of x is b, so b = k + 3.
Coefficient of constant term is c, so c = 3k - 220.
Solve these equations to find the values of k, b, and c.
Substitute the value of k in the second equation: b = -27 + 3.
Simplify it further: b = -24.
Now substitute the value of k in the third equation: c = 3(-27) - 220.
Simplify it further: c = -81 - 220 = -301.
So, the values of a, b, and c are: a = -27, b = -24, and c = -301.
(B) To find the solutions of the equation f(x) = 0, substitute the values of a, b, and c in the given polynomial equation f(x) = 6x^3 - ax^2 + bx + c and then solve for x.
f(x) = 6x^3 + 27x^2 - 24x - 301.
Unfortunately, due to the complexity of the equation, the solutions cannot be obtained without further information or the use of numerical methods.