Question

is there a way to tell if a binomial divides evenly into a polynomial with out dividing everything out??

Answer 1

try wolframalpha.com that helped me with that.

Answer 2

Are you familiar with synthetic division? That's the method I'd use to determine whether or not a given binomial divides into a given polynomial with a zero remainder. If you want examples, let me know.

Answer 3

We haven't yet covered that, could I get some examples?

Answer 4

Google it. Its much easier to do after seeing a few examples.

Answer 5

f g is a zero of ax^3+bx^2+cx+d then (x-g) divides the polynomial.

Answer 6

Please give me an example problem from your homework (a polynomial and the binomial that we're to test).

Answer 7

for instance: (x^3 - x^2 -9x + 14) / (x-2)
it asks for you to "state of the given binomial is a factor of the given polynomial"

Answer 8

*if

Answer 9

that possible factor (x-2) translates into divisor 2.
That 3rd order poly yields four coefficients:

1 -1 -9 14.

Set up synthetic division as follows:

2 | 1 -1 -9 14

---------------------
1 (bring down that coefficient, 1)

Multiply that 1 at the bottom by the divisor, 2, obtaining 2, and write this product (2) under the -1:

2 | 1 -1 -9 14
2
---------------------
1 1 Now multiply the 2nd ' 1 ' at the bottom by the divisor (2), obtaining product 2, and write that product under the -9.

2 | 1 -1 -9 14
2 2
---------------------
1 1 -7 Multiply that -7 by the divisor (2). Write
the product under that 14. Combine that 14 and -14. What do you get?

Answer 10

0, which means that there is no remainder which in turn means that x-2 goes into the polynomial evenly which means it is a factor, so the answer is yes?

Answer 11

that's Right! zero remainder indicates that (x-2) is a factor and 2 is a root of the given polynomial. Try anotehr problem on your own; share your results with me if you're unsure.

Answer 12

one question, why is the divisor 2 and not a -2?

Answer 13

do you reverse the sign in front of the number? so binomial (x+7) would translate to a divisor of -7?

Answer 14

Exactly! That's a keen observation. YES!

Answer 15

Great! Thank you so much, it's actually much simpler than I first believed it to be, again, thanks! Also just as a quick check to make sure that I am doing this right, would the answer to the equation (4a^3 + 27a^2 -14a - 49) / (x+7) be yes?

Answer 16

also what if your divisor has a number in front of the x? for instance... (5x + 4) would the divisor just be 4? or..

Answer 17

Very, very good question! I would set 5x+4= to 0 and solve for x; the result, x=-4/5, would be the divisor to use in synth. div. Nice work!

Answer 18

would the answer to the equation (4a^3 + 27a^2 -14a - 49) / (x+7) be yes?

Give me just a moment. I see that the divisor in synth. div. would be -7.

Answer 19

oh! so that is why you reverse the sign, because you are solving for the zeros of the equation?

Answer 20

Yes, yes, and yes.

(x+7) is indeed a factor (with no remainder) of that polynomial you gave me.

Answer 21

Nice work!

Answer 22

another question, if I am given a problem with the values out of order, do I need to correct them in order to start the division process? for instance if I had something that looked like 16x + 4x^5 - 4x^2 - 44p^3 + 8 - 14x^4 also if there were to be no x^4 would i have to put in 0 as a place holder?