Question

what is factor theorem and remainder theorem

Answer 1

FACTOR THEOREM - in algebra factor theorem is a theorem link factors and zeroes of a polynomial . it states that apolynomial has a factor if and only if it is a root.
REMAINDER THEREM - remainder theorem is an application of euclidean division of polynomial . it states that remainder of division of polynomial is equal to.

Answer 2

Factor theorm
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let f (x) b a polynomial. if polynomial f (x) is Dvided by (x = C), then the remainder will be zero .
that is x= c is zero or root of a polynomial f (x) which also makes(x- c) is a factor of f (x).
does the theorem states that if f(x) is equal to zero then (x - 3) is a factor of the polynomial f (x) the Converse of this theorem is also true. that is if (x -c) is a factor of the polynomial f x then f (c) = zero.

Proof
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consider a polynomial f (x) which is divided by (x-c)

then f (c) = 0

thus by the remainder theorm
f (x)= (x-c)q (x) + f (c)
=(x-c)q (x)+0
=(x-c)q

thus (x-c) is a factor of polynomial f (x).

Remainder theorm.
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consider the polynomial g(x) of any degree greater than or equal to 1 .and any real number see if g(x) is divided by a linear polynomial (x -C),then remainder is equal to g(c).

that is,

g(x)=(x-c)q (x) + g (c)

Proof
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consider the polynomial g (x) is devided by (x-c).

let q (x) be quotient and R be the remainder then by using the division algorithm

divided = (division × quotient) + remainder.

g (x)= {(x-c) × q(x)} + R

g (c)= {(c-c) × q(c)} + R

g (c)= {0 × q(c) + R

thus g(c)= R

hence the remainder is g (c).