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

Find the remainder when $large 4x^3-12x^2+x-2$ is divided by 2x +1
A. -9	B. 0	C. -5	D. -6
Right Option : D
View Explanation

Question : 2

What is the remainder when $large 4x^3+8x^2+4x+2$ is divided by x+2 ?
A. -8	B. 10	C. 16	D. -6
Right Option : D
View Explanation

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