To determine the values of B, R(x), and Q(x) that make the equation true, we need to manipulate the equation and compare the coefficients of like terms on both sides.
The given equation is:
3x^3 + 5/x + 1 = Ax^2 + Bx + C + R(x)/Q(x)
First, let's focus on the left side of the equation:
3x^3 + 5/x + 1
To simplify the expression, we can find a common denominator for the terms. The common denominator is x, so we can rewrite the equation as:
(3x^4 + 5 + x)/x
Now, we have the left side in a single fraction. To make it match the form on the right side, which is Ax^2 + Bx + C + R(x)/Q(x), we can equate the numerators and denominators of the two sides.
Comparing the numerators:
3x^4 + 5 + x = Ax^2 + Bx + C + R(x)
Comparing the denominators:
x = Q(x)
From the comparison, we can deduce the following values:
B = 1 (coefficient of x)
R(x) = 5 (constant term)
Q(x) = x (denominator)
Therefore, the values that make the equation true are B = 1, R(x) = 5, and Q(x) = x.