Question

Factor theorem
Let P(x) be a polynomial

Prove that
a.)If P(c)= 0 then x-c is a factor of P(x)
b.) If x-c is a factor of P(x then P(c)= 0

Answer 1

@ganeshie8 pls help

Answer 2

@phi pls help

Answer 3

according to http://www.purplemath.com/modules/factrthm.htm
the "remainder theorem" states that you can write a polynomial p(x) as
p(x) = (x – a)q(x) + r(x)

If x – a is indeed a factor of p(x), then the remainder after division by x – a will be zero. That is:

p(x) = (x – a)q(x)

The factor theorem states that if you get a zero remainder (r(x) up above is 0) then (x-a) is a factor of p(x)

Answer 4

for
a) let p(x)= 0
factor p(x) into its factors (x-a1)(x-a2)...(x-an) so we have
(x-a1)(x-a2)...(x-an) =0
the solution are the n roots x= a1, x= a2, .... x= an
that means if p(c)=0 then c is one of these n roots, say c= a1
which also means (x-c) is a factor of p(x)

b) If x-c is a factor of P(x) then P(c)= 0

Use the remainder theorem to write P(x)= (x-c)Q(x) + R(x)
Now, by the factor theorem, if x-c is a factor of P(x), then R(x) is zero, therefor
P(x)= (x-c)Q(x)

if x= c then (x-c)=0 and P(c)= 0 * Q(x)= 0