Question

Explain And Prove Factor Theorem

Answer 1

Step-by-step explanation:

FACTOR THEOREM

Let f(x) be a polynomial of degree n>1 and let a be any real number.

1) If (a) = 0 then (x-a) is a factor of f(x).

2) If (x-a) is a factor of f(x) then f(a) = 0.

PROOF

1) Let f(a) = 0

On dividing f(x) by (x-a) ,let g(x) be the quotient.

Also,by the remainder theorem,when f(x) is divided by ( x-a), then the remainder is f(a).

Therefore , f(x) = (x-a) •g(x) + f(a)

» f(x) = (x-a) •g(x) [ f(a) = 0 given]

» ( x-a) is a factor of f(x).

2) Let (x-a) be a factor of f(x).

On dividing f(x) by (x-a), let g(x) be the quotient.

Then,f(x) = (x-a) •g(x)

» f(a) = 0 [ putting x= a]

Thus, (x-a) is a factor of f (x) » f(a) = 0.

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