Question

Help with finding how many factors?

Answer 1

as it says, use the factor theorem ?

Answer 2

if \(x-k\) is a factor of \(f\) then \(f(k)=0\)... so to determine whether \(x-3\) is a factor merely evalaute \(f(3)\) and see if it's zero

Answer 3

similarly for \(x+4=x-(-4)\) you evaluate \(f(-4)\) and see if it's zero

Answer 4

for \(4x+3\) observe that \(4x+3=4(x+3/4)\) so just focus on the \(x+3/4\) part and evaluate \(f(-3/4)\)

Answer 5

Okay, that makes sense : )

Answer 6

Thanks!

Answer 7

no problem!!! so for example:$$f(3)=4(3)^4-13(3)^3-80(3)^2+189(3)+180=324-351-720+567+180=0$$therefore \(x-3\) is indeed a factor of \(f\)

Answer 8

remember what it means to be a factor ... if \(x-k\) is a factor of \(f\) that means \(f\) can be written as \(f(x)=(x-k)(\dots)\) i.e. the product of \(x-k\) and some other junk.

watch what happens when we plug in \(k\):$$f(x)=(x-k)(\dots)\\f(k)=(k-k)(\dots)\\f(k)=0\cdot(....)=0$$since \(0\) times any number is \(0\) :-)

therefore if \(x-k\) is a factor then it must be that \(f(k)=0\)

Answer 9

wut

Answer 10

Okay thank you so much for all the help! =)