Question

I don’t understand :(

Answer 1

The quotient when
`x^3-4x^2-8x+8` is divided by
`x+2` is
`x^2-6x+4` with no remainder, so x+2 is a factor of p(x).

Explanation:

As the question suggests, there are two ways to solve this problem: long division (which can always be used to divide polynomials), and synthetic division (which can only be used when you are dividing by something of the form
`(x-a)`. In this case, we may use synthetic division, since
`x+2` is equal to
`x-(-2)`. I will use synthetic division here as it is slightly faster than long division.

The -2 on the left represents that we are dividing by x-(-2), and the rest of the numbers in the top row are the coefficients of p(x): 1, -4, -8, and 8.

The first step in synthetic division is to being down the leading coefficient of the polynomial: in this case, the 1 (as indicated in red). Now, we multiply the 1 by -2. We get -2, which we place directly below the -4.

Next, we add directly down the column, (-4 + (-2) is equal to -6), and this answer is placed in the box below. We can continue this process, getting the coefficients 1, -6, 4, and 0 in the bottom row.

This is the answer: the quotient is
`x^2-6x+4`, and the remainder is zero (which indicates that x+2 is a factor of p(x)).

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Edit: long division

We may also solve this problem using long division (see the second image). The first step is to look at the leading coefficients: since
`x * x^2` is equal to
`x^3`, the first term in the quotient will be
`x^2`. Since
`x^2 * (x+2)` is equal to
`x^3+2x^2`, we must now subtract that from
`x^3-4x^2-8x+8`.

We repeat the same process, as shown in the image. Since we eventually get to zero, the remainder is zero, and the polynomial at the top (x^2-6x+4) is the quotient.

I know this might be difficult to follow, so please comment if you have any questions.