Question

Q. What is difference between Remainder Theorem and Factor Theorem ?

Give few examples .

Answer 1

hi buddy here is ur answer

u might know the division rule that is

dividend = divisor x quotient + remainder

same is the remainder theorem but it is slightly different instead of numeric values we have an expression

remainder theorem states that

let p(x) be any polynomial of degree greater than or equal to one and also let a be any real number

if p(x) is divided by any linear polynomial [having degree one ] (x-a) then the remainder is p(a)

example : p(x) = 3x^4-4x^3-3x-1 when divided by (x-1)

the remainder is p(1)

substitute i in place of x

we get ,p(1) = 3(1)^4 - 4(1)^3-3(1) -1

= 3-4-3-1

= p(1) = -5

now comes the factor theorem

it states that if p(x) is a of degree greater than 1 and a is any real number then x-a is a factor of p(x)

if p(a) = 0 then x-a is a factor of p(x)

example : p(x) = x^3 + 3x^2 + 5x +6 when divided by x +2

then zero of polynomial x+2 = -2

p(-2) = (-2)^3 + 3(-2)^2 + 5(-2) +6

= -8 +12-10+6

p(-2)=0

hence x+2 is a factor of p(x)

hope it helps you buddy !!!