Perform the indicated operations on the following polynomials.Divide: 6x3 + 27x - 19x2 - 15 by 3x -5

EXPLANATION:
Given;
We are given the polynomial below;
[tex]6x^3+27x-19x^2-15[/tex]Required;
We are required to divide the polynomial by;
[tex]3x-5[/tex]Step-by-step solution;
We shall apply the synthetic division method of dividing polynomials.
The first step would be to re-arrange the polynomial in standard form. This is shown below;
[tex]6x^3-19x^2+27x-15[/tex]Next step, we list out the coefficients of the polynomial;
[tex]6,-19,27,-15[/tex]Next step, we identify the zeros of the denominator;
[tex]\begin{gathered} 3x-5=0 \\ \\ 3x=5 \\ \\ x=\frac{5}{3} \end{gathered}[/tex]We can now write down the question in synthetic division format;
Next step, we carry down the leading coefficient below the division symbol.
Next step, we multiply this value by the zero of the denominator that is, 5/3.
That gives us;
[tex]6\times\frac{5}{3}=10[/tex]Now we write 10 right under the next coefficient and that is -19. We add both together (-19 + 10 = -9) and write the result below the division symbol. Next we multiply this too by the zero of the denominator and we have;
[tex]-9\times\frac{5}{3}=-15[/tex]We write this too under the next coefficient and we have;
[tex]27-15=12[/tex]We multiply this too by 5/3 and we have 20. Write this right under the next coefficient and add up and we now have;
[tex]-15+20=5[/tex]The result we have come up with are the coefficients beneath the division symbol and that is;
[tex]6,-9,12,5[/tex]The last number is the remainder and the result of the division carried out will be;
[tex]6x^2-9x+12\text{ }Rem\text{ }5[/tex]This is otherwise written out as follows;
ANSWER:
[tex]6x^2-9x+12+\frac{5}{(3x-5)}[/tex]