5x^(3)+ 6x^(2)+8 =(Ax+1)(x-2)(x+B)+x-5C
To find the values of A, B, and C in the equation 5x^3 + 6x^2 + 8 = (Ax + 1)(x - 2)(x + B) + x - 5C, we will expand the right side of the equation and compare it with the left side.
First, let's expand the right side using the distributive property:
(Ax + 1)(x - 2)(x + B) + x - 5C
= (Ax^2 + (A - 2) x - 2Ax + x - 2)(x + B) + x - 5C
= (Ax^2 + (A - 2) x - 2Ax + x - 2) x + (Ax^2 + (A - 2) x - 2Ax + x - 2) B + x - 5C
= Ax^3 + (A - 2) x^2 - 2Ax^2 + x^2 - 2x + Ax - 2x + 2 + Bx - 2B + x - 5C
= Ax^3 - Ax^2 + x^2 + (A - 2 + 1) x^2 - 2x + Ax + x + Bx + 1x - 2x + 2 - 2B - 5C
= Ax^3 + (A - 1) x^2 + (A + 4)x + 2 + Bx + x - 2B - 5C
Now, we can compare the expanded right side with the left side, 5x^3 + 6x^2 + 8:
Ax^3 + (A - 1) x^2 + (A + 4)x + 2 + Bx + x - 2B - 5C = 5x^3 + 6x^2 + 8
By comparing the coefficients of like terms on both sides, we can determine the values of A, B, and C.
Comparing the coefficients:
Coefficient of x^3: A = 5 (since the coefficient of x^3 on the left side is 5x^3)
Coefficient of x^2: A - 1 = 6 (since the coefficient of x^2 on the left side is 6x^2, then A - 1 must be 6)
Coefficient of x: A + 4 = 1 (since the coefficient of x on the left side is 1x, then A + 4 must be 1)
Constant term: 2 - 2B - 5C = 8 (since the constant term on the left side is 8)
From the second equation, A - 1 = 6, we can solve for A:
A - 1 = 6
A = 6 + 1
A = 7
From the third equation, A + 4 = 1, we can solve for A:
A + 4 = 1
A = 1 - 4
A = -3
However, we have a contradiction with the values obtained for A. Therefore, there is no consistent solution for A, B, and C that satisfies the given equation.