Answer:
[tex]f(x) = 2x^2 - x - 1[/tex]
[tex]g(x) = 2x + 1[/tex]
Step-by-step explanation:
Let the two polynomials be represented with f(x) and g(x);
Since, we have to generate the polynomial ourselves;
I'll make use of the following:
[tex]f(x) = 2x^2 - x - 1[/tex]
[tex]g(x) = 2x + 1[/tex]
Note that; when the result of polynomial division is referred to as quotient;
To get the quotient, we simply divide f(x) by g(x)
[tex]\frac{f(x)}{g(x)} = \frac{2x^2 - x - 1}{2x + 1}[/tex]
Expand the numerator
[tex]\frac{f(x)}{g(x)} = \frac{2x^2 - 2x + x - 1}{2x + 1}[/tex]
[tex]\frac{f(x)}{g(x)} = \frac{2x(x - 1) +1(x - 1)}{2x + 1}[/tex]
Factorize:
[tex]\frac{f(x)}{g(x)} = \frac{(2x + 1)(x - 1)}{2x + 1}[/tex]
Cross out 2x + 1
[tex]\frac{f(x)}{g(x)} = x - 1[/tex]
This implies that, the quotient, Q(x) is
[tex]Q(x) = x - 1[/tex]
Comparing the divisor g(x) and the quotient Q(x), we notice that they have the same degree of 1
