Question

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

Answer 1

EXPLANATION:

Given;

We are given the polynomial below;

$6x^3+27x-19x^2-15$

Required;

We are required to divide the polynomial by;

$3x-5$

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;

$6x^3-19x^2+27x-15$

Next step, we list out the coefficients of the polynomial;

$6,-19,27,-15$

Next step, we identify the zeros of the denominator;

$\begin{gathered} 3x-5=0 \\ \\ 3x=5 \\ \\ x=\frac{5}{3} \end{gathered}$

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;

$6\times\frac{5}{3}=10$

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;

$-9\times\frac{5}{3}=-15$

We write this too under the next coefficient and we have;

$27-15=12$

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;

$-15+20=5$

The result we have come up with are the coefficients beneath the division symbol and that is;

$6,-9,12,5$

The last number is the remainder and the result of the division carried out will be;

$6x^2-9x+12\text{ }Rem\text{ }5$

This is otherwise written out as follows;

ANSWER:

$6x^2-9x+12+\frac{5}{(3x-5)}$