SOLUTION: Can you please help me solve this? {{{(16x^5 - 32x^4 - 81x + 162) / (x - 2)}}} I am supposed to solve and state whether the binomial is a factor of the polynomial. I used synthe

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Question 583689: Can you please help me solve this? I am supposed to solve and state whether the binomial is a factor of the polynomial.
I used synthetic division to drop the coefficients and got:
2] 16 -32 0 0 -81 162
0 32 0 0 0 -162
_______________________
16 0 0 0 0 -81 0
(The zeroes in the top row are because there are no x-cubed and x-squared variables, therefore they're respresented by a zero, right?)
Multiple choice allowed me the answers of:
A - 1348; no
B - 0; yes
C - 700; yes
D - 0; no
I am pretty sure the answer is C because my remainder is zero, and x-2 is a factor of the original equation. If this is not right, can you please show me what I've done wrong?
(Also, I apologize if the synthetic division does not line up properly when submitted and if it makes it harder for you to read.)
Found 3 solutions by richwmiller, solver91311, Edwin McCravy:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
If your remainder is zero then why isn't it B.
(16x^5-32x^4-81x+162)/(x-2) does equal 16x^4-81 which can be factored to
(-3+2x)*(3+2x)*(9+4x^2)
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


No. Not answer C.

The numeric part of the answer is supposed to be the remainder that results from the synthetic division which is the same as the value of the function evaluated at the divisor. That is, if your function is and the synthetic division divisor is because you are testing as a factor of , then the rightmost number in the third row of numbers in your synthetic division is equal to . And since is a factor of if and only if , your rightmost result is, in fact, 0, that means B is your answer. Answers C and D are impossible because, in the case of C, the number cannot be non-zero and the answer yes, and in the case of D, the number cannot be zero and the answer no.

John

My calculator said it, I believe it, that settles it
The Out Campaign: Scarlet Letter of Atheism
Answer by Edwin McCravy(20086)   (Show Source): You can put this solution on YOUR website!


The numerator can be factored by grouping; no synthetic division required:



Factor 16x4 out of the first two terms and -81 out of the last two terms:



Now factor (x-2) out of the two terms in the numerator:



That's enough to show that the binomial x-2 is a factor of numerator and
denominator.  However let's finish factoring. We can factor  as
the difference of squares:



We can further factor  as the difference of squares:



And as long as x isn't 2, we can cancel the (x-2)'s and get



Edwin