Respuesta :
Using polynomial division, it is found that the expression that represents the amount of candy sold each year per type at Cassandra's Candy Corner is:
[tex]2x^2 - 5x + 52[/tex]
The amount of candy sold per type is given by the following division:
[tex]\frac{C(x)}{T(x)} = \frac{4x^3 + 10x^2 + 54x + 520}{2x + 10}[/tex]
The denominator can be written as:
[tex]2x + 10 = 2(x + 5)[/tex]
At the numerator, to see if we can simplify, we verify if x = -5 is a factor:
[tex]C(-5) = 4(-5)^3 + 10(-5)^2 + 54(-5) + 520 = 0[/tex]
Since C(-5) = 0, it is a factor of the numerator, and thus, since the numerator is of the 3rd degree, it can be written as a 3 - 1 = 2nd degree polynomial multiplying x + 5:
[tex](ax^2 + bx + c)(x + 5) = 4x^3 + 10x^2 + 54x + 520[/tex]
[tex]ax^3 + (5a + b)x^2 + (5b + c)x + 5c = 4x^3 + 10x^2 + 54x + 520[/tex]
Equaling both sides:
[tex]a = 4[/tex]
[tex]b = 10 - 5a = -10[/tex]
[tex]5c = 520 \rightarrow c = 104[/tex]
Thus:
[tex]C(x) = (4x^2 - 10x + 104)(x + 5)[/tex]
And:
[tex]\frac{C(x)}{T(x)} = \frac{(4x^2 - 10x + 104)(x + 5)}{2(x + 5)} = 2x^2 - 5x + 52[/tex]
Thus, the expression is:
[tex]2x^2 - 5x + 52[/tex]
A similar problem is given at https://brainly.com/question/13586325
