Remainder Theorem

If p(x) and g(x) are two polynomials in which the degree of p(x)\ge degree of g(x) and  g(x) \ne 0 are given then we can get the  q(x) and r(x) so that:

 P(x) = g(x)\ q(x) + r(x),

where  r(x) = 0 or degree of  r(x) < degree of  g(x).

It says that  p(x) divided by g(x) , gives q(x) as quotient and  r(x) as remainder.

Let’s understand it with an example

Division of a Polynomial with a Monomial

3x^3+x^2+x\div x=\dfrac{3x^3}{x}+\dfrac{x^2}{x}+\dfrac{x}{x}=3x^2+x+1

We can see that ‘x’ is common in the above polynomial, so we can write it as

3x^3+x^2+x=x(3x^2+x+1)

Hence 3x^2+x+1 and x the factors of  3x^3 + x^2 + x.

Steps of the Division of a Polynomial with a Non –Zero Polynomial

Divide x^2 - 3x -10 by 2 + x

Step 1:  Write the dividend and divisor in the descending order i.e. in the standard form. x^2 - 3x -10 and x + 2

Divide the first term of the dividend with the first term of the divisor.

\dfrac{x^2}{x}=x this will be the first term of the quotient.

Step 2: Now multiply the divisor with this term of the quotient and subtract it from the dividend.

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Step 3: Now the remainder is our new dividend so we will repeat the process again by dividing the dividend with the divisor.

Step 4:  \left(\dfrac{5x}{x}\right)=-5

Step 5: 

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The remainder is zero.

Hence x^2 - 3x - 10 = (x + 2)(x - 5) + 0

Dividend = (Divisor \times Quotient) + Remainder

Remainder Theorem says that if p(x) is any polynomial of degree greater than or equal to one and let ‘t’ be any real number and p (x) is divided by the linear polynomial x-t , then the remainder is p(t) .

As we know that

P(x) = g(x)\ q(x) + r(x)

If p(x) is divided by (x-t) then

If x=t

P (t) = (t - t).q (t) + r = 0

To find the remainder or to check the multiple of the polynomial we can use the remainder theorem.

Example:

What is the remainder if a4 + a3 – 2a2 + a + 1 is divided by a – 1.

Solution:

P(x) = a4 + a3 – 2a2 + a + 1

To find the zero of the (a – 1) we need to equate it to zero.

a -1 = 0

a = 1

p (1) = (1)4 + (1)3 – 2(1)2 + (1) + 1

= 1 + 1 – 2 + 1 + 1

= 2

So by using the remainder theorem, we can easily find the remainder after the division of polynomial.

Factor Theorem

Factor theorem says that if p(y) is a polynomial with degree n \ge 1 and t is a real number, then

  1. (y – t) is a factor of p(y), if p(t) = 0, and
  2. P (t) = 0 if (y – t) is a factor of p(y).

Example: 1

Check whether  g(x) = x - 3 is the factor of  p(x) = x^3- 4x^2+ x + 6  using factor theorem.

Solution:

According to the factor theorem if  x - 3 is the factor of p(x) then  p(3)=0, as the root of  x - 3 is 3.

 P (3) = (3)^3- 4(3)^2 + (3) + 6

= 27 -36 + 3 + 6 = 0

Hence, g(x) is the factor of  p(x).

Example: 2

Find the value of k, if x-1 is a factor of p(x)=kx^2-\sqrt{2}x+1

Solution:

As x-1 is the factor so p(1)=0

p(x)=kx^2-\sqrt2x+1

P(1)=k(1)^2-\sqrt2(1)+1=0

=k-\sqrt2+1=0

=k=\sqrt2-1

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