How to Find remainder of any Polynomial in 2 easy steps ?

      For a long time I wanted to write about remainder theorem, as most of my students used to ask me frequently about this and I thought probably you also want to read about this topic. So, here it is...

Remainder theorem

     As I've have discussed about Polynomial in my earlier post and also about Long Division method to solve any fractions of polynomials, Now this time is to explain the most favourable method to find remainders of fractions.

     With the help of long division method remainder and quotient can be evaluated but this turn to be lengthy method, while we are only tend to find, whether any fractions are divisible or not.

      

     So, when someone is looking to find whether the Polynomial is perfectly divisible by another polynomial or not i.e. remainder theorem comes in existence to find remainders direct without any hassle and struggle with only in two steps.

    

 • So here is step by step method of remainder theorem:-

      Let any quadratic, cubic or any Polynomial P(x) is divided by another polynomial say g (x) then remainder can be evaluated by following method of two easy steps

Say P(x) = x³ + 3x² - 5x +6 

  &    g(x) = x + 1

• STEP 1:  by g(x) = 0,

               x+1= 0

               x = -1

• STEP 2:  P (-1) = (-1)³ + 3 (-1)² - 5(-1) + 6

                   = -1     + 3 ×1    + 5   + 6

                   = -1  + 3 + 11

        P (-1)  = 13

hence, If P(x) = x³ + 3x² - 5x +6 when divided by g(x) = x + 1 left with remainder 13.

the same remainder can be obtained from long Division method and can verify this.

let's have an another example:

P(x) = x³ + 3x² + 3x + 1

& g(x) = 5+2x

• STEP 1:   5+2x = 0  

               2x = -5

              x =  -5/2

 now, 

• STEP 2:  

   P(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1

             = -125/8  + 75/4  - 15/2 + 1

             = -27/8

     hence, I hope you've got to know about remainder theorem and it's most utility in the field of algebra.

 I personally consider this method is a shortcut to verify whether any Solution of  Long Division is correct or not by observing remainders.
now over to you, what you have found about this tutorial special and what else you want to add here do let me know by commenting.

How to find square roots without long Division method

I wanted to share this post from a long time but somehow this couldn't be possible, but now here it is,

How to find square roots without long Division method

                                      

yes..!!

There is one method i.e. approximation method but this wouldn't​ lead us towards exact solution at all time,

 " Confuse...??"

don't worry see this example.

Q. Find √23

Sol. using approx. method 

      A perfect square before 23 is 16  and just next perfect square is 25.

so this can be written as 

 
 √16 < √23 < √ 25

      4  <  4.7or 4.8 < 5 ( this is an approximation)

but  √23 = 4.79583

 • And now our short method to evaluate square root of √23...

Next highest square root is √25 

so,

let 23 = A

      25 = B

• our formula of finding square root of will be 

 •   √A = √ B + (A - B)/ 2×√B

 √23   = 5    + (23-25)/ 2× 5

          = 5    - 2/ 10 

          = 5   - 0.2

          = 4.8

 •    let's have an another example for √45
so,
45 = A
49 = B
putting values into Formula

√A =  √ B + (A - B)/ 2×√B
√45 = 7  +  ( 45-49) /2×7
       =  7 -  4/14
       =  7 - 0.28571
       =  6.71429      

now by evaluating from calculating
√45 = 6.7082 ≈ 6.71429

so this Method is giving us nearly equals to exact values.

   • No doubts this method  is easy than long Division and also giving answer to satisfactory manner, But it requires​ practice to become perfect. Try at least for twice or thrice with different values.

So this is our today'as tutorial of finding square root of any number without our traditional method.
Have a look at some best tutorials From Math's Buddies:

• Math's Magic Square

• Euclid's Math for polyhedron

Now over to you, let me know how you have found this tutorial helpful to you, by commenting.

3 simple easy steps to solve long Division

I've already explained some of the basic and elementary topics related to polynomials (for the topic click here)  and here in this space one more topic is remaining i.e. long Division.

Long Division of polynomials

If a polynomial p(x) is divided by another polynomial say g (x) and quotient as q(x) and remainder as r(x) then these set of polynomials can be represented as Euclid's Division Algorithm i.e.

  •  p(x) = q(x) × g(x) + r(x)

According to our topic we are to divide p(x) by g(x) so,

  •   p(x) / g(x) = q(x)   if remainder becomes 0.

if remainder doesn't becomes 0 then above relationship can be rearranged as 

  •   [p(x) - r(x) ] / g(x) = q(x)

for practice let's take an example:

in this above image an example is taken in which  p(x) = x² + 7x +12 is divided by 

g(x)= x+3 and resultant quotient is q(x)= x+4.

Steps for long Division method -

• make the same term- take a look at the first term of both dividend and Divisor, to male the same term  of g(x) as p(x)'s  first term we should choose a suitable term. In above example x is chosen so that when it is multiplied by x ( g(x)'s first term) it will give us x² (p(x)'s first term).

• Do not forget- yes...!!! of course this step can't be missed out, as most of the students makes mistake by forgetting other terms of g(x) which should also be multiplied and resultant should placed under p(x)'s other respective terms.In above example x of quotient is multiplied by 3 and 3x is placed under 7x because of like terms.

• Subtract and proceed- now after multiplying all the terms move onto subtraction as normal subtraction.In above example 4x + 12 is remaining after subtracting, after doing this, proceed the same process until we get either remainder 0 or degree of remainder less than that of g(x).

So these are three simple easy steps which I found to be remarkable for your solutions of same problems.

Now check out an another example-

If still you are finding difficulties feel free to comment under the post in comment box.

read out our featured post :

• Solving Quadratic equation by MTS

Euclid's Division Algorithm:Algebra form

Euclid's Division Algorithm

is also have it's utility in algebra in same formation.

whenever, one intended to perform long Division of algebraic terms then that procedure also follow Euclid Division Algorithm, or I can give you an idea to clear the concept so have a look at this,

when we divide 23 by 5
then this divided as
23/5 = 4 ³/5

here quotient is 4 and remainder is 3.
But to justify this division we tend to multiply quotient to Divisor and result is added to remainder hence
5×4+ 3 =23

this is what can be expressed in generalisation as

B×q+ r = A

What Euclid has given.

On the other aspect of algebra this can be applied similarly likewise.

But the representation is quiet different as each term is considered as polynomial in algebra and represented by P(x) that's why
p(x) is considered as Dividend
q(x) as quotient
r(x) as remainder
and
g(x) as Divisor

so the algorithm will be

p(x) = q(x) × g(x) + r(x)

this is for justification of long Division of algebraic terms.
Alternatively this formula is useful to evaluate any of the polynomial be it quotient​, be it remainder or Divisor​ if all the other three polynomials are given.
Suppose, we've to evaluate q(x), then

q(x) =  [ p(x) - r(x) ]/ g(x).
Now, over to you let me know by comment if I'm missing something.