Factor And Remainder Theorem


 
 
Concept Explanation
 

Factor And Remainder Theorem

Value of a Polynomial:

The value of a polynomial f(x) at x = a is obtained by substituting the value of x= a in the given polynomial and it is denoted by f(a)

Illustration: Find the value of polynomial f(x) at x = 1, x= 1/2

f(x)= 8x^2-3x+7

f(1)= 8(1)^2-3(1)+7 = 8-3+7 = 12

f(frac{1}{2})= 8(frac{1}{2})^2-3(frac{1}{2})+7

         = 8(frac{1}{4})-3(frac{1}{2})+7

        = (frac{8}{4})-(frac{3}{2})+7

        = (frac{8-6+28}{4})

        = (frac{30}{4})=frac{15}{2}

Zero of a Polynomial:

The value of x for which the value of the polynomial becomes zero is called zero of polynomial.

Illustration: The value of the polynomial at x= 1

f(x)= x^3-6x^2+11x-6

f(1)= (1)^3-6(1)^2+11(1)-6

          = 1-6(1)+11(1)-6

          = 1-6+11-6 =0

As f(1) is zero so x=1 is the zero of the polynomial

and (x-1) is a factor of this polynomial.

Factor and Remainder Theorem:

It states that:

For a polynomial p(x) of degree greater than of equal to one and 'a' be any real number. If p(x) is divided by the linear polynomial x-a then the remainder is p(a)

Illustration: Find the remainder when f(x) is divided by x-1

f(x)= x^4+x^3-2x^2+x+1

and the zero of x-1 is 1

f(1)= (1)^4+(1)^3-2(1)^2+(1)+1

          = 1+1-2+1+1=2

So by the Remainder Theorem the remainder when f(x) is divided by x-1  is 2

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Complete the following statements with an appropriate word/term to be filled in the blank spaces (s).

x - a is a factors of the polynomial p(x), if p(a) = __________

 

Right Option : C
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Explanation
Question : 2

What is the remainder when large 4x^3+8x^2+4x+2 is divided by x+2 ?

Right Option : D
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Explanation
Question : 3

Which of the following are the zeroes of the polynomial p(x)=2x^2-3x+1 ?

Right Option : A
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Explanation
 
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