NCERT Class 9 Maths Solutions Chapter - 2 Polynomials, Ex 2.5

Ex - 2.5

Question 1.  Use suitable identities to find the following products:

Solution

(i).    By using identity 

 

        

(ii).    By using identity 

 

         

 

(iii).    

 

 

           By using the identity  

           

(iv).    By using identity 

 

          

 

(v).    By using identity 

 

         

 

 

Concept Insight: If the value of the two terms of the binomials are equal then use the algebraic identity (x+a) (x-a) = x² - a² else use (x+a) (x+b) = x²+(a+b)x+ab to obtain required product.

Question 2.  Evaluate the following products without multiplying directly:

    (i)    103  107        (ii)    95  96    (iii)    104  96

Solution

(i).    103  107 = (100 + 3) (100 + 7)
                          = (100)² + (3 + 7) 100 + (3) (7)
        [By using the identity, , where

        x = 100, a = 3 and b = 7]
                          = 10000 + 1000 + 21 
                          = 11021

(ii).   95  96 = (100 - 5) (100 - 4)
                    = (100)² + (- 5 - 4) 100 + (- 5) (- 4) 
        [By using the identity, , where

        x = 100, a = - 5 and b = - 4]
                    = 10000 - 900 + 20
                    = 9120

(iii).  104  96 = (100 + 4) (100 - 4)
                    = (100)² - (4)²           

                    = 10000 - 16
                    = 9984

 

Concept Insight: The key is to use the algebraic identity (x+a) (x+b) = x²+(a+b)x+ab or (x+a) (x-a) = x² - a²  for such questions. Write each of the numeral as  100 � k ,  or any other suitable number whose square can be easily computed.

Question 3.  Factorise the following using appropriate identities:

Solution

Concept Insight: Use the appropriate square identity.  If the polynomial has only two terms, reduce each term to the perfect square and use the algebraic identity . When the polynomial has three terms and the term having

 

unit power of each variable has negative sign use the square identity  else use  .

Question 4.  Expand each of the following, using suitable identities:

Solution

Concept Insight: Use the algebraic identity  

 

. Do consider the sign of terms while multiplying and squaring.

Question 5.  

Solution

Concept Insight: Use the algebraic identity  

 

  in the reverse order. Write each term as per the terms of the standard identity. Do consider the sign of terms involved.

Question 6.  

Solution

Concept Insight: Since the expressions involves cube so cubic identity will be used.  If the terms of the given polynomial are separated by positive sign use the identity   or if  negative signs are used  then use . Carefully apply the mathematical operations.

Question 7.  Evaluate the following using suitable identities:

(i).  (99)³        (ii).  (102)³        (iii).  (998)³

^Solution

We know that

 

 

(i)    (99)³ = (100 - 1)³
                = (100)³ - (1)³ - 3(100) (1) (100 - 1)
                = 1000000 - 1 - 300(99)
                = 1000000 - 1 - 29700
                = 970299

(ii)    (102)³ = (100 + 2)³
                  = (100)³ + (2)³ + 3(100) (2) (100 + 2)
                  = 1000000 + 8 + 600 (102)
                  = 1000000 + 8 + 61200
                  = 1061208 


(iii)    (998)³ = (1000 - 2)³
                   = (1000)³ - (2)³ - 3(1000) (2) (1000 - 2)
                   = 1000000000 - 8 - 6000(998)
                   = 1000000000 - 8 - 5988000
                   = 1000000000 - 5988008
                   = 994011992

 

 

Concept Insight: Use the cubic identity  and  . Write the numerical term as something added or

 

subtracted from 10,100, 1000 or higher powers of 10 as it's easy to compute higher powers of 10. Carefully apply the mathematical operations.

Question 8.  

Solution

Concept Insight: Since all the polynomial given here have degree 3 so cubic identities would be used here. Now if all the terms of the given polynomial are positive  then use identity   while if any two terms has negative sign reduce each of

 

the term of the polynomial as per the standard cubic identity  . 

Question 9.  

Solution

Concept Insight: When the two terms of the polynomial are separated by positive sign use the identity   and when by negative sign use .

 

Carefully take the common term out.

Question 10.  

Solution

Concept Insight: Reduce the terms of the polynomial to perfect cube and then if the two terms of the polynomial are separated by positive sign use the identity   and when by negative sign use  .

Question 11.  Factorise: 27x³ + y³ + z³ -9xyz

Solution

We Know that

        

 

Concept Insight: Reduce each terms of the polynomial as per the left hand side of the standard identity,   .

Question 12.  

Solution

We know that

 

                         

 

Concept Insight: Since the left hand side of the identity resembles the left hand side of identity,  , so this identity

 

will be applicable here. Now the right hand side of the above identity can be written into many forms we need to look at what is required to proved, Accordingly apply  mathematical simplifications and square identities to get the desired result.

Question 13.  If x + y + z = 0, show that x³ + y³ + z³ = 3xyz

Solution

Concept Insight: Use the result that    for x + y + z = 0.

Question 14.  Without actually calculating the cubes, find the value of each of the following:

Solution

Concept Insight: Use the result   since x + y + z = 0. Also consider the

 

sign of the term. Carefully do the computation.

Question 15.  Give possible expressions for the length and breadth of each of the following rectangle, in which their areas are given:

Solution

We know that, 
 Area = length  breadth

 

Concept Insight: For such questions factorise the expression, given for the area of rectangle by splitting the middle term. One of its factors will be its length and the other will be its breadth.

Question 16.  What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

Solution

We know that, 
Volume of cuboid = length  breadth  height

(i).    

        So, the possible solutions is
        Length = 3, breadth = x, height = x - 4

 

        

 

Concept Insight: For such questions factorise the expression, given for the volume of the cuboid by taking the common term out if it has two terms and by splitting the middle term if the polynomial has three terms. Three factors obtained will be its length  breadth and  height.