Question

How do I figure this problem out without a calculator?

Answer 1

Which of the following is a factor of \[x^{17}-4x {^{15}}-x^3+4\]
A.) x+2
B.) x-2
C.) x+1
D.) x-1
E.) x-0

Answer 2

when the sum of ALL the co-efficients of a polynomial is = 0
then x= 1 is the roo, x-1 is the factor

Answer 3

root*

Answer 4

when the sum of the co-efficients of even powers of x
= the sum of the co-efficients of odd powers of x

then x=-1 is the root and x+1 is the factor

Answer 5

sum of all the co-efficients in
\(x^{17}-4x {^{15}}-x^3+4\)
is
1-4-1+4 = ... ?

Answer 6

the sum would be 0. But how do you get x=-1?

Answer 7

sum of all co-efficients =0
so x-1 is the factor.

now how we find whether x +1 is the factor?

example:
\(x^2+2x+1\)
odd powers of x : 2x
even powers of x : x^2, 1

sum of co-efficient of odd powers of x : 2
sum of co-efficient of even powers of x : 1+1 = 2

so x+1 is the factor

Answer 8

in your case

odd powers of x : x^17 , -4 x^15, -x^3
even powers of x : +4

sum of co-efficient of odd powers of x : 1 -4 -1 = -4
sum of co-efficient of even powers of x : 4

so x+1 is NOT the factor

Answer 9

making any sense?

Answer 10

Yeah I get it now. Thank you!

Answer 11

welcome ^_^