We want to take the quotient of two given polynomials.
The result of the division is:
p(x) = 2*x^2 - 9*x + 10
We want to solve:
[tex]\frac{2*x^3 - 3*x^2 -17*x + 30}{x + 3}[/tex]
First, notice that -3 is a zero of the denominator, then we can write the denominator as:
p(x)*(x + 3) = 2*x^3 - 3*x^2 -17*x + 30
Where p(x) is a polynomial of second degree and the solution of our division.
We can write:
p(x) = a*x^2 + b*x + c
Then:
p(x)*(x + 3) = 2*x^3 - 3*x^2 -17*x + 30
( a*x^2 + b*x + c)*(x + 3) = 2*x^3 - 3*x^2 -17*x + 30
a*x^3 + b*x^2 + c*x + 3*a*x^2 + 3*b*x + 3*c = 2*x^3 - 3*x^2 -17*x + 30
a*x^3 + (b + 3*a)*x^2 + (3*b + c)*x + (3*c) = 2*x^3 - 3*x^2 -17*x + 30
All the coefficients must be equal in both sides, so we get:
a = 2
b + 3*a = -3
3*b + c = -17
3*c = 30
replacing the first equation into the second one we get:
b + 3*2 = -3
b + 6 = -3
b = -3 - 6 = -9
From the last equation we get:
3*c = 30
c = 30/3 = 10
Then we got:
a = 2
b = -9
c = 10
then the result of the division is:
p(x) = 2*x^2 - 9*x + 10
If you want to learn more, you can read:
https://brainly.com/question/11536910
