OpenStudy (anonymous):

Help and explain "Polynomial Long Division" ?: 3x+1 / 3x^3 - 5x^2 + 10x -3

4 years ago
OpenStudy (whpalmer4):

It really is very similar to doing long division with numbers. For a first step in this problem, for example, you would ask yourself what is the greatest factor you could multiply with 3x+1 to get an expression starting with 3x^3? Multiply 3x+1 by that factor, subtract the result from your dividend, and repeat, now with a polynomial whose highest degree is x^2. It looks a little weird sometimes, because you can get terms appearing where no term was before (dividing a polynomial with terms that have coefficients of 0, such as x^3 + 2x (but no x^2 term)), but you just have to have faith :-)

4 years ago
OpenStudy (zehanz):

I am used to this notation: 3x+1 / 3x^3 - 5x^2 + 10x -3 \ ...... The answer will be on the dots. Here we go: Look at the 3x of 3x+1 and at 3x³ of that other polynomial. If you multiply 3x with x², you get 3x³, so multiply the whole of 3x+1 with x² and put the result below and subtract: (btw: put the x² on the right). Drop 10x. 3x+1 / 3x³ - 5x² + 10x -3 \ x² 3x³ + x² - -------- -6x² + 10x Now look again: how to multiply 3x with a number to get -6x²? You need -2x. So do the same as before: 3x+1 / 3x³ - 5x² + 10x -3 \ x² - 2x 3x³ + x² - -------- -6x² + 10x -6x² - 2x - ---------- 8x - 3 Now once again: multiply 3x with a number that will make it 8x, so that is 8/3: 3x+1 / 3x³ - 5x² + 10x -3 \ x² - 2x + 8/3 3x³ + x² - -------- -6x² + 10x -6x² - 2x - ---------- 8x - 3 8x +8/3 - -------- -5 2/3 The remainder is -5 2/3

4 years ago
OpenStudy (zehanz):

The outcome of (3x³-5x²+10x-3)/(3x+1) is x²-2x+8/2 - (5 2/3)/(3x+1). In an ideal situation the remainder is 0 and everybody is happy :D

4 years ago
OpenStudy (anonymous):

Thankyou guys so much! I'm understanding it a little bit! :) confusing.

4 years ago
OpenStudy (whpalmer4):

It does take some practice, and don't forget when you are subtracting polynomials to watch out for - -...

4 years ago
OpenStudy (anonymous):

So do I change the signs if it's a negative?

4 years ago
OpenStudy (zehanz):

Just try a simpler one (that will have a remainder 0): Divide x³+6x²+11x+6 by x+2: I'll help you to get started: x+2 / x³+6x²+11x+6 \ x² x³ +... - -----

4 years ago
OpenStudy (whpalmer4):

The tricky part is if you're dividing by something with a negative sign in it, like (3x-1) instead of (3x+1). I like ZeHanz's way of writing the subtraction symbol on the line rather than in front of the quantity being subtracted, because just sticking a - out in front won't do the right thing, it is really - (stuff) which means that any negative terms in (stuff) are actually added. I apologize in advance if I confuse matters here by giving you an example, but I'm about to leave... 3x-1 / 3x^3 - 13x^2 + 16x - 4 \ x^2 3x^3 - x^2 <- x^2(3x-1) = 3x^3-x^2 - ----------- -12x^2 + 16x <- -13x^2 -(-x^2) = -13x^2 + x^2 = -12x^2 3x-1 / 3x^3 - 13x^2 + 16x - 4 \ x^2 - 4x 3x^3 - x^2 - ----------- -12x^2 + 16x -12x^2 + 4x - -------------- 12x - 4 3x-1 / 3x^3 - 13x^2 + 16x - 4 \ x^2 - 4x + 4 3x^3 - x^2 - ----------- -12x^2 + 16x -12x^2 + 4x - -------------- 12x - 4 12x - 4 - -------- 0

4 years ago
OpenStudy (zehanz):

It is all about carefully looking at the several "-" signs and concentration :)

4 years ago
OpenStudy (whpalmer4):

I actually multiply by - (whatever my term in the quotient is), and then add, just to avoid having to do the double negatives. My example would look like 3x-1 / 3x^3 - 13x^2 + 16x - 4 \ x^2 -3x^3 + x^2 + ----------- -12x^2 + 16x etc.

4 years ago
OpenStudy (anonymous):

The problem you did whpalmaer4 definitely helped alot, I rewrote both of you guys problems in my notebook, and practiced with them. I get confused with the negative signs, but I also remember to separate it from the problem. Thankyou! :)

4 years ago
OpenStudy (anonymous):

Oh, and what if the problem is like x^2 - 56 Don't you replace it with zeros, or no?

4 years ago
OpenStudy (whpalmer4):

Yes, x^2 - 56 would be written out as x^2 - 0x - 56 Don't forget that you can always check your work (and get a bit of polynomial multiplication practice) by multiplying the quotient you get with the divisor and making sure you get the dividend back.

4 years ago
OpenStudy (zehanz):

And practise..(a lot)

4 years ago