Dividing Poly nomials

Hello!
My dd needs help dividing Polynomials, any one out there know how to do this? Its Alg 2 aka SOS11 and MUS alg2

 

here i found this... it makes sense to me but it took a bit to remember

There are two cases for dividing polynomials: either the "division" is really just a simplification and you're just reducting a fraction, or else you need to do long polynomial division (which is covered on the next page).

* Simplify (2x + 4)/2

This is just a simplification problem, because there is only one term in the polynomial that you're dividing by. And, in this case, there is a common factor in the numerator (top) and denominator (bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split the division into two fractions, each with only one term on top, and then reduce:

2x/2 + 4/2 = x + 2

...or else I can factor out the common factor from the top and bottom, and then cancel off:

2(x + 2)/2 = x + 2

Either way, the answer is the same: x + 2

* Simplify (21x^3 - 35x^2) / (7x)

Again, I can solve this in either of two ways: by splitting up the sum and simplifying each fraction separately: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

21x^3/7x - 25x^2/7x = 3x^2 - 5x

...or else by taking the common factor out front and canceling it off:

7x(3x^2 - 5x)/7x = 3x^2 - 5x

Either way, the answer is the same: 3x2 – 5x

* Simplify [ x(x + 3) - 2(x + 3) ] / (x + 3)

I can split the sum and reduce each fraction separately:

x(x + 3)/(x + 3) - 2(x + 3)/(x + 3) = x - 2

The numerator (top) does indeed have a common factor; it's just a rather large one. Since both terms contain the factor "x + 3", then this is a common factor, and may be factored out front:

(x + 3)(x - 2)/(x + 3) = x - 2

Either way, the answer is the same: x – 2

 

Polynomial Long Division (page 2 of 3)

Sections: Simplification and reduction, Polynomial long division

If you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

* Divide x2 – 9x – 10 by x + 1

Think back to when you were doing long division with plain old numbers. You would be given one number that you had to divide into another number. You set up the division symbol, inserted the two numbers where they belonged, and then started making guesses. And you didn't guess the whole answer right away; instead, you started working on the "front" part (the larger place values) of the number you were dividing. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Long division for polynomials works in much the same way:
First, I set up the division:

For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leading x2 of the dividend.
Set up the division
If I divide the leading x2 inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top:

Put the 'x' up top
Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by the x (on the "side"), and carry the x2 underneath:
Carry the 'x^2' down
Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath:
Carry the '1x
Then I'll draw the "equals" bar, so I can do the subtraction.

To subtract the polynomials, I change all the signs in the second line...




Change signs

...and then I add down. The first term (the x2) will cancel out:


Subtract


I need to remember to carry down that last term, the "subtract ten", from the dividend:


Carry down the '–10'
Now I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the –10x by the x, I would end up with a –10, so I'll put that on top:
Put '–10' up top



Now I'll multiply the –10 (on top) by the leading x (on the "side"), and carry the –10x to the bottom:


Carry the '–10x' down


...and I'll multiply the –10 (on top) by the 1 (on the "side"), and carry the –10 to the bottom:


Carry the '–10' down


I draw the equals bar, and change the signs on all the terms in the bottom row:


Change the signs



Then I add down:




Subtract

Then the solution to this division is: x – 10

Since the remainder on this division was zero (that is, since there wasn't anything left over), the division came out "even". When you do regular division with numbers and the division comes out even, it means that the number you divided by is a factor of the number you're dividing. For instance, if you divide 50 by 10, the answer will be a nice neat "5" with a zero remainder, because 10 is a factor of 50. In the case of the above polynomial division, the zero remainder tells us that x + 1 is a factor of x2 – 9x – 10, which you can confirm by factoring the original quadratic dividend, x2 – 9x – 10.

 

Polynomial Long Division: Examples (page 3 of 3)

Sections: Simplification and reduction, Polynomial long division

* Simplify (x^2 + 9x + 1)/(x + 7)

This can be done in either of two ways: I can factor the quadratic and then cancel the common factor, like this:

(x + 2)(x + 7)/(x + 7) = x + 2

But what if I didn't know how to factor? I can always use long division:

animation

(I mustn't forget to change my signs, as shown in red, when I'm doing the subtraction.)

The answer to the division is quotient, the polynomial across the top: x + 2

* Divide 3x3 – 5x2 + 10x – 3 by 3x + 1

animation

This division did not come out even. What am I supposed to do with the remainder?

Think back to when you did long division with plain numbers. Sometimes there would be a remainder; for instance, if you divide 132 by 5:

long division

...there is a remainder of 2. Remember how you handled that? You made a fraction, putting the remainder on top of the divisor, and wrote the answer as "twenty-six and two-fifths":

132/5 = 26 + 2/5

The first form, without the "plus" in the middle, is how "mixed numbers" are written, but the meaning of the mixed number is actually the addition.

We do the same thing with polynomial division. Since the remainder is –7 and since the divisor is 3x + 1, then I'll turn the remainder into a fraction (the remainder divided by the original divisor), and add this fraction to the polynomial across the top of the division symbol. Since the division looks like this:

animation

...then the answer is this: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

x^2 - 2x + 4 + (-7)/(3x + 1)

Warning: Do not write the polynomial "mixed number" in the same format as numerical mixed numbers! If you just append the fractional part to the polynomial part, this will be interpreted as polynomial multiplication, which is not what you mean!

Note: Different books format the long division differently. When writing the expressions across the top of the division, some books will put the terms above the same-degree term, rather than above the term being worked on. In such a text, the long division above would be presented as shown here:

The only difference is that the terms across the top are shifted to the right. Otherwise, everything is exactly the same. You should probably use the formatting that your instructor uses.






same as the animation above, but with x^2, -2x, and +4 shifted one place to the right

* Divide 2x3 – 9x2 + 15 by 2x – 5

First off, I note that there is a gap in the degrees of the terms of the dividend: the polynomial
2x3 – 9x2 + 15 has no x term. My work could get very messy inside the division symbol, so it is important that I leave space for a x-term column, just in case. I can create this space by turning the dividend into 2x3 – 9x2 + 0x + 15. This is a legitimate mathematical step: since I've only added zero, I haven't actually changed the value of anything.

Now that I have all the "room" I might need for my work, I'll do the division:

animation

I need to remember to add the remainder to the polynomial part of the answer:

x^2 - 2x - 5 + (-10)/(2x - 5)

* Divide 4x4 + 3x3 + 2x + 1 by x2 + x + 2

I'll add a 0x2 term to the dividend (inside the division symbol) to make space for my work, and then I'll do the division in the usual manner:

animation

Then my answer is:

4x^2 - x - 7 + (11x + 15)/(x^2 + x + 2)

To succeed with polyomial long division, you need to write neatly, remember to change your signs when you're subtracting, and work carefully, keeping your columns lined up properly. If you do this, then these exercises should not be very hard; annoying, maybe, but not hard.

 

Here's a great explanation:

http://www.youtube.com/watch?v=4u8_AMacu-Y&feature=youtube_gdata

 

Thanks! She watched the videos and read this thread and I think ithelped! She walked away with a smile! Thanks so much!