Question

Use the long division method to find the result when 3x³ + 11x² - 12x - 19 isdivided by 3x + 2. If there is a remainder, express the result in the formr(x)q(x) + b(z)

Answer 1

Given two polynomials f(x) and g(x), the division between them can be represented by:

`(f(x))/(g(x))=q(x)+(r(x))/(g(x))`

Where q(x) is the quotient and r(x) is the remainder. From the problem, we identify:

`\begin{gathered} f(x)=3x^3+11x^2-12x-19 \\ \\ g(x)=3x+2 \end{gathered}`

Then, using the long division method:

Finally:

`\begin{gathered} r(x)=-7 \\ q(x)=x^2+3x-6 \end{gathered}`

And the result is:

`(f(x))/(g(x))=x^2+3x-6-(7)/(3x+2)`