Question

Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of the polynomial being divided.

Answer 1

If after dividing the polynomial by the binomial, the remainder is 'zilch' or zero, then the binomial is a factor of the polynomial, otherwise not.

Answer 2

i need an example of both

Answer 3

choose a binomial. Choose some other polynomial. Multiply together. The result will have the binomial as a factor. There's one example. For the second, choose a binomial. Choose some kind of wacky polynomial. It probably won't be divisible by the binomial. There's your two examples.

Answer 4

Hmm okay I 'll give you an example of one, try to guess how you 'll find the other one where there is a remainder left.

So let me take a polynomial, say = (x^2-4)(x+3), which simplifies to x^3+3x^2-4x-12
Now ofcourse, since the above polynomial's constituent factors are (x^2-4) and (x+3), both these binomials will divide it completely.

So, the polynomial could be x^3+3x^2-4x-12, and for the binomial divisor I'll take, say, (x+3).

Okay so now, try to guess how do I make the polynomial term above NOT divisible by (x+3).

Answer 5

hmm im still not really understanding.

Answer 6

okay i 'll give you the simplest possible example.

let me take the no. 6.
what are its factors?

Answer 7

3 and 2

Answer 8

very well.
now, what does that mean?

that 2 and 3 each divides 6 completely, without leaving a remainder right?

Answer 9

yes

Answer 10

i get the first one im just not getting how to make it not divisble

Answer 11

now suppose i choose a no. 3x4. which equals 12. obviously 3 and 4 will be factors of it. right? divide it completely that means.

Answer 12

Okay, hmm
so back to the first example then

how would you make 6 not divisible by 3?

Answer 13

i mean what changes would you make to the no. to make that happen.

Answer 14

change it ot 7?

Answer 15

That's right. You added something that was not divisible by 3.

Answer 16

See if you can do the same thing with polynomials... they're kind of just numbers anyway.

Answer 17

do i just take out a a number and add a new one?

Answer 18

do the same you did with the numbers before. You had a number that was divisible by some number. You wanted to add something to the first so it was no longer divisible by the second. You added something that wasn't divisible by the second (you added 1).

You have a polynomial divisble by some other polynomial. You want to make it not divisible. So you add something not divisible by that divisor polynomial.

Answer 19

yes @haileystowers exactly. you added 1 to 6, made it 7. Now it's not divisible by 3

But what if you added 3 to it? Then it would become 9, and again become divisible by 3.

So the key to remember is that, when you are making it indivisible by a factor, rememeber not to add any multiple of that factor, or it would again become divisible by that factor.


In the example I posted, (x^3+3x^2-4x-12). is divisible by (x+3)

So I can add anything like 2, 3, 2x+1 or the sort... but not anything like x+3, or 2(x+3)...


so I 'll add 2 to the polynomial, make it x^3+3x^2-4x-10, and that, is indivisible by x+3



Hope it's clear now.