how to divide this using polynomial long division?

When dividing polynomials take the divisors first term, and divide by the dividends first term. You know (hopefully), that x+3 is the divisor, just as in 9 / 3, 3 is the divisor. So, we take the first term x^3 of the dividend, and divide the first term of the divisor (x^3) by it. We get x^2. This is the FIRST part of our answer, (it will be the first term of the quotient).

Now, we take the newly acquired first term of the quotient, and multiply it by the divisor. So x^2 (x + 3) = x^3 + 3x^2

Now, we subtract this from the dividend, to get a new dividend. So x^3 + x^2 - 2x - (x^3 + 3x^2) = -2x^2 - 2x

Repeat this process on the new dividend. We have -2x^2 / x (remember, first term of NEW dividend, divided by the first term of the divisor, which has not changed). We get -2x.

This 2x is the SECOND term of our new quotient. So we have x^2 - 2x as our quotient so far. Once again multiply this second term of the new quotient by the divisor: -2x (x + 3) = -2x^2 - 6x. And subtract this from the NEW dividend, which was -2x^2 - 2x:

So -2x^2 - 2x - (-2x^2 - 6x) = -2x^2 - 2x + 2x^2 + 6x = 4x

So far: quotient = x^2 - 2x, new dividend = 4x

Perform the final calculation:

You'll divide 4x (from the new dividend) by x, and get 4. This is the almost final term of out quotient. You'll see why it's almost the final term. So our quotient now is x^2 - 2x + 4

Multiply 4 by (x + 3) = 4x + 12, and subtract this from the new dividend (4x)

We get 4x + 12 - (4x + 12) = 0

So we've divided until we've reached 0, this 0 means there is no remainder! Awesome :)

(Sorry if I left out the 12 in some of the earlier calculations, it really just gets in the way)

So our final quotient is: x^2 - 2x + 4

We can check this by performing: (x^2 -2x + 4)(x + 3)

Why? Because we perform a / b = c. In this case a was the original polynomial. b was the divisor (x + 3), and c is our newly acquired quotient. Now, this means that b (a / b) = c (b) = a = cb. So if we multiply the quotient by the divisor, we SHOULD get our original polynomial. Let's check:

(x^2 -2x + 4)(x + 3) = x^3 - 2x^2 + 4x + 3x^2 -6x + 12 = x^3 + x^2 -2x + 12

Awesome, we got the correct answer.

The main steps are:

1. Divide first term of dividend, by the divisor.
2. Place this new term in the quotient.
3. Multiply that term by the divisor.
4. Subtract that result from the dividend to create a new dividend.
5. Repeat steps 1 - 4 on the new dividend.
NOTE: IF YOU EVER get a first term in the dividend that is SMALLER than the
first term of the divisor, STOP, and place whatever the dividend is over the divisor and add it to the quotient, it is the remainder.

Ex: If you attempted to perform 12 / (x + 3), You'd see that you can't perform 12 / x. So in this case you simply place the dividend (12), over the divisor (x+3). Obviously you get what you started with, but that's the point, you can't divide 12 / (x + 3) to make it any simpler, it IS a "remainder" so to speak.

Polynomial long division just takes practice but you'll get it eventually, just do a few examples. It also looks MUCH nicer on paper than it does on Y!A

Needs parentheses: (x³ + x² - 2x + 12) ÷ (x+3)

Same steps as regular long division
x³ ÷ x = x²
subtract x²(x+3) from x³ + x² - 2x + 12
Etc.
https://www.flickr.com/photos/dwread/26020247183/in/dateposted-public/

x + 3 = x - (-3)

-3 I 1_____1_____-2_____12
_ I_____ -3______6____ -12
_ I 1____-2______4_____ 0

x² - 2x + 4