Question

I have a question about polynomial division it is in the picture

Answer 1

To answer this question, we have to do the long division process for polynomials. We can do the operation as follows:

To do this division process, we have:

1. Divide the first term of the dividend by the first element of the divisor. They are:

$\frac{-4x^3}{4x^2}=-x$

2. Now, we have to multiply this result by the divisor, and the result will change its sign since we have to subtract that result from the dividend as follows:

$-x\cdot(4x^2_{}-4x-4)=-4x^3+4x^2+4x$

And since we to subtract this result from the dividend, we end up with:

$-(-4x^3+4x^2+4x)=4x^3-4x^2-4x$

3. Then we have the following algebraic addition:

$\frac{\begin{cases}-4x^3+24x^2-15x-15 \\ 4x^3-4x^2-4x\end{cases}}{20x^2-19x-15}$

4. Again, we need to divide the first term of the dividend by the first term of the divisor as follows:

$\frac{20x^2}{4x^2}=5$

5. And we have to multiply 5 by the divisor, and the result will be subtracted from the dividend:

$5\cdot(4x^2-4x-4)=20x^2-20x-20$

Since we have to subtract this from the dividend, we have:

$-(20x^2-20x-20)=-20x^2+20x+20$

6. And we have to add this algebraically to the dividend we got in the previous step:

$\frac{\begin{cases}20x^2-19x-15 \\ -20x^2+20x+20\end{cases}}{x+5}$

And this is the remainder of the division, x + 5.

As we can see from the division process, we got as:

1. The quotient: -x + 5

$q=-x+5$

2. The remainder: x + 5.

$R=x+5$

Since we have that the dividend = divisor * quotient + remainder.

Therefore, the result for this division is:

$-4x^3+24x^2-15x-15=(4x^2-4x-4)\cdot(-x+5)+(x+5)$