8th Class Mathematics Algebraic Expressions Multiplication and Division of Algebraic Expressions

         Multiplication and Division of Algebraic Expressions  

 

Various Algebraic Relations to be used in this chapter

1.             \[{{(A+B)}^{2}}={{A}^{2}}+{{B}^{2}}+2AB\]

2.             \[{{(A-B)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\]

3.             \[{{A}^{2}}-{{B}^{2}}=(A-B)(A+B)\]

4.             \[{{(A+B+C)}^{2}}={{A}^{2}}+{{B}^{2}}+{{C}^{2}}+2AB+2BC+2CA\]

5.             \[{{A}^{3}}+{{B}^{3}}+{{C}^{3}}-3ABC=(A+B+C)({{A}^{2}}+{{B}^{2}}+{{C}^{2}}\]\[-AB-BC-CA)\]

6.             \[{{(A+B)}^{3}}={{A}^{3}}+{{B}^{3}}+3{{A}^{2}}B+3A{{B}^{2}}\]

7.             \[{{(A-B)}^{3}}={{A}^{3}}-{{B}^{3}}-3{{A}^{2}}B+3A{{B}^{2}}\]

8.             \[{{(A+B)}^{4}}={{A}^{4}}+4{{A}^{3}}B+6{{A}^{2}}{{B}^{2}}+4A{{B}^{3}}+{{B}^{4}}\]

9.             \[{{(A-B)}^{4}}={{A}^{4}}-4{{A}^{3}}B+6{{A}^{2}}{{B}^{2}}-4A{{B}^{3}}+{{B}^{4}}\]

10.          \[{{A}^{3}}-{{B}^{3}}=(A+B)({{A}^{2}}+{{B}^{2}}-AB)\]

11.          \[{{A}^{3}}-{{B}^{3}}=(A-B)({{A}^{2}}+{{B}^{2}}+AB)\]    

 

 

 

 

Find the product of \[(2{{x}^{2}}-5x+4)\] and \[({{x}^{2}}+7x-8)\]

(a) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}+68x+32)\]            

(b) \[(2{{x}^{4}}+9{{x}^{3}}-47{{x}^{2}}+68x-32)\]

(c) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}+68x+32)\]

(d) \[(2{{x}^{4}}-9{{x}^{3}}-47{{x}^{2}}-68x-32)\]

(e) None of these  

 

Answer: (b)    

 

 

 

 

The product of \[(3x+5y)\] and \[(5x-7y)\] is.

(a) \[15{{x}^{2}}+4xy-35{{y}^{2}}\]          

(b) \[5{{x}^{2}}-4xy+35{{y}^{2}}\]

(c) \[5{{x}^{5}}+4xy+35{{y}^{2}}\]           

(d) \[{{x}^{2}}-4xy-35y\]

(e) None of these  

 

Answer: (a)

Explanation:

\[=(3x+5y)(5x-7y)\]

\[=3x(5x-7y)+5y(5x-7y)\]

\[=15{{x}^{2}}-21xy+25xy-35{{y}^{2}}\]

\[=15{{x}^{2}}+4xy-35{{y}^{2}}\]  

 

 

 

 

The product of \[(-3{{x}^{2}}y)(4{{x}^{2}}y-3x{{y}^{2}}+4x-5y)\] is .........

(a) \[12{{x}^{3}}{{y}^{2}}+9{{x}^{2}}{{y}^{2}}-12{{x}^{4}}y+15{{x}^{3}}{{y}^{3}}\]

(b) \[-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{3}}y+15{{x}^{2}}{{y}^{2}}\]

(c) \[-12x{{y}^{2}}+9{{x}^{3}}{{y}^{2}}-12{{x}^{3}}y+15xy\]

(d) \[-12x{{y}^{3}}+9xy-12x{{y}^{2}}+15{{x}^{2}}y\]

(e) None of these  

 

Answer: (b)

Explanation:

\[(-3{{x}^{2}}y)(4{{x}^{2}}y-3x{{y}^{2}}+4x-5y)=\]\[-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{4}}{{y}^{2}}+9{{x}^{3}}{{y}^{3}}-12{{x}^{3}}y+15{{x}^{2}}{{y}^{2}}\]

Therefore, option (b) is correct and rest of options is incorrect.  

 

 

 

 

The product of \[(3{{x}^{2}}+{{y}^{2}})\] and \[(2{{x}^{2}}+3{{y}^{2}})\] is ------.

(a) \[{{x}^{3}}+10{{x}^{2}}{{y}^{3}}+3{{y}^{4}}\]                               

(b) \[3{{x}^{2}}+5{{x}^{5}}{{y}^{2}}+3{{y}^{4}}\]

(c) \[6{{x}^{5}}+10{{x}^{2}}{{y}^{3}}+3{{y}^{4}}\]                             

(d)\[6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]

(e) None of these  

 

Answer: (d)

Explanation:

\[(3{{x}^{2}}+{{y}^{2}})(2{{x}^{2}}+3{{y}^{2}})\]

\[=6{{x}^{4}}+9{{x}^{2}}{{y}^{2}}+2{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]

\[=6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+30{{y}^{4}}\]  

 

 

            Problems Based on Identities  

 

  Multiply: \[(3x+2y)(3x+2y)\]

(a) \[9{{x}^{2}}+4{{y}^{2}}+12xy\]                           

(b) \[18{{x}^{3}}+2{{y}^{2}}+10xy\]

(c) \[9{{x}^{2}}+4{{y}^{2}}+8xy\]                              

(d) \[9{{x}^{2}}+6{{y}^{3}}+12xy\]        

(e) None of these

 

Answer: (a)

Explanation:

\[(3x+2y)+(3x+2y)=3x(3x+2y)+2y(3xx+2y)\]

\[9{{x}^{2}}+6xy+6xy+4{{y}^{2}}=9{{x}^{2}}+12xy+4{{y}^{2}}\]  

 

 

  Solve: \[(4{{x}^{2}}+5)(4{{x}^{2}}+5)\]

(a) \[16{{x}^{4}}+25+40{{x}^{2}}\]                           

(b) \[16{{x}^{4}}+28+30{{x}^{2}}\]

(c) \[16{{x}^{4}}+30+20{{x}^{2}}\]                           

(d) \[16{{x}^{4}}+8+25{{x}^{2}}\]

(e) None of these

 

Answer: (a)

Explanation:

\[(4{{x}^{2}}+5)(4{{x}^{2}}+5)={{(4{{x}^{2}}+5)}^{2}}\]

\[{{(4{{x}^{2}})}^{2}}={{5}^{2}}+2(4{{x}^{2}})\times 5\] [Using\[~{{(\text{a}+\text{b})}^{\text{2}}}={{\text{a}}^{\text{2}}}+{{\text{b}}^{\text{2}}}+\text{2ab}\]]

\[=16{{x}^{4}}+25+40{{x}^{2}}\]

Multiplication of both the expressions is as same as option (a).  

 

 

  \[(4x-7y)(4x-7y)\] equal to:

(a) \[16{{x}^{2}}+56xy+49{{y}^{2}}\]                      

(b) \[16{{x}^{2}}-8xy+49{{y}^{2}}\]

(c) \[16{{x}^{2}}-56xy-49{{y}^{2}}\]                         

(d)\[16{{x}^{2}}-56xy+49{{y}^{2}}\]

(e) None of these

 

Answer: (d)

Explanation:

\[(4x-7y)(4x-7y)=(4{{x}^{2}})-2.4x.7y+{{(7y)}^{2}}=\]\[16{{x}^{2}}=56xy+49{{y}^{2}}\]  

 

 

  Solve the expression: \[\left( x-\frac{3}{x} \right)\left( x-\frac{3}{x} \right)=\]

(a) \[{{x}^{2}}+\frac{9}{{{x}^{2}}}-18\]                                   

(b) \[{{x}^{3}}+\frac{9}{{{x}^{2}}}-6\]

(c) \[{{x}^{2}}+\frac{9}{{{x}^{2}}}-6\]                                     

(d) \[{{x}^{2}}+\frac{1}{{{x}^{2}}}-6\]

(e) None of these

 

Answer: (c)

Explanation:

\[\left( x-\frac{3}{x} \right)\left( x-\frac{3}{x} \right)={{x}^{2}}{{\left( \frac{3}{x} \right)}^{2}}-2\times x\times \frac{3}{x}={{x}^{2}}+\frac{9}{{{x}^{2}}}-6\]  

 

 

  Simplify: \[(4x+5y)(4x-5y)\]

(a) \[16{{x}^{2}}-25xy\]                                

(b) \[16{{x}^{2}}-5{{y}^{2}}\]

(c) \[16{{x}^{2}}-25{{y}^{2}}\]                    

(d) \[16{{x}^{2}}-{{y}^{2}}\]

(e) None of these

 

Answer: (c)

Explanation:

\[(4x+5y)(4x-5y)={{(4x)}^{2}}-{{(-5y)}^{2}}=16{{x}^{2}}-25{{y}^{2}}\]  

 

 

  Simplify: \[(2x+3y)(2x-3y)\]

(a) \[{{x}^{2}}-3{{y}^{2}}\]                                          

(b) \[2{{x}^{2}}-3{{y}^{2}}\]

(c) \[4{{x}^{2}}-9{{y}^{2}}\]                                        

(d) \[{{x}^{2}}-9{{y}^{2}}\]

(e) None of these

 

Answer: (c)

Explanation:

\[(2x+3y)(2x-3y)={{(2x)}^{2}}-{{(3y)}^{2}}=4{{x}^{2}}-9{{y}^{2}}\]    

 

 

 

 

• An expression with one or more terms is called a polynomial.
• It can be used to find the sum of any order of the polynomials.
• A Strange Prime Number - The prime number 73,939,133 has a very strange; property. If you keep removing a digit from the right hand end of the number, each of the remaining numbers is also prime.   
• At the rate of moving 1 disc per second around the clock, it is estimated to take about 585 billion years to complete the transfer of the gold discs in the tower of Brahma.             

 

 

 

 

• While adding or subtracting the algebraic expression we always add or subtract; the like terms.             
• While multiplying we multiply each term of the expression with each term of the other expression.    
• The constants multiplied with the variables are called the coefficient of the term.
• If the variables and powers of the terms are same then they are called like terms otherwise they are called unlike terms.
• An identity is an equality, which is true for all values of the variables in the equality.                
• An algebraic expression consists of variables and constants.    

 

 

 

  Find the sum of the given algebraic expression. \[2{{p}^{2}}{{q}^{2}}-3pq+4,4+7{{p}^{2}}{{q}^{2}}-8pq,\]\[5{{p}^{2}}{{q}^{2}}-8+9pq\]

(a) \[1{{p}^{2}}{{q}^{2}}-5pq+5\]                             

(b) \[14{{p}^{2}}{{q}^{2}}-2pq+1\]

(c) \[11{{p}^{2}}{{q}^{2}}+5pq+10\]                        

(d)\[14{{p}^{2}}{{q}^{2}}+19pq+17\]

(e) None of these  

 

Answer: (b)  

 

 

  Find the sum of the given expressions: \[-5{{x}^{2}}+3x-8,\] \[4x+7x-2{{x}^{2}}\] and \[6-2x+2{{x}^{2}}\]

(a) 2x+3y+z                                        

(b) x+2y+3z

(c) 9x+6y+4z                                      

(d) -5x2 + 5x + 5

(e) None of these  

 

Answer: (d)

Explanation:

Writing the given expression in descending powers of\[x\]in the form of rows with the like terms as shown below and adding them column wise we get,

\[\begin{align}   & -5{{x}^{2}}+3x-8 \\  & -2{{x}^{2}}+4x+7 \\    & \,\,\,\underline{2{{x}^{2}}-2x+6} \\  & -\underline{5{{x}^{2}}+5x+5} \\ \end{align}\]  

 

 

  Subtract: \[\text{3a}-\text{3b}+\text{c}\] from \[\text{4a}+\text{5b}-\text{3c}\]

(a) \[\text{2a}+\text{3b}+\text{c}\]                       

(b) \[\text{a}+\text{2b}+\text{3c}\]

(c) \[\text{a}+\text{8b}-\text{4c}\]                                        

(d)\[-5{{x}^{2}}+5x+5\]

(e) None of these  

 

Answer: (c)

Explanation:

\[\begin{align}   & \,\,\,\,\,\text{4a}+\text{5b}-\text{3c} \\  & \pm \,\underline{\text{3a }+\text{ 3b }\pm \text{ c}} \\  & \,\,\,\,\underline{\text{a }+\text{ 8b}-\text{4c}} \\ \end{align}\]  

 

 

  What must be subtracted form \[3{{a}^{2}}-6ab-3{{b}^{2}}-1\] to get \[4{{a}^{2}}-7ab-4{{b}^{2}}+1\]?

(a) \[-{{a}^{2}}+ab+{{b}^{2}}-2\]                              

(b) \[{{a}^{2}}+ab+{{b}^{2}}+2\]

(c) \[{{a}^{2}}-ab-{{b}^{2}}+2\]                                 

(d) \[{{a}^{2}}-ab-4{{b}^{2}}-2\]

(e) None of these  

 

Answer: (a)

Explanation:

\[\begin{align}   & \text{3}{{\text{a}}^{\text{2}}}-\text{6ab}-\text{3}{{\text{b}}^{\text{2}}}-1 \\  & \underline{\pm \text{4}{{\text{a}}^{\text{2}}}\mp \text{7ab}\mp 4{{\text{b}}^{\text{2}}}\pm 1} \\  & \,\,\,-{{a}^{2}}+ab+{{b}^{2}}-2 \\ \end{align}\]  

 

 

  Find the product of \[5{{m}^{2}}n,-3mnp\] and \[-5{{n}^{2}}p\]

(a) \[75{{m}^{3}}{{n}^{3}}{{p}^{2}}\]                      

(b) \[75{{m}^{2}}{{n}^{4}}{{p}^{2}}\]

(c) \[75m{{n}^{3}}{{p}^{3}}\]                                     

(d) \[75{{m}^{3}}n{{p}^{2}}\]

(e) None of these  

 

Answer: (b)

Explanation:

\[5{{m}^{2}}n\times (-3mnp)\times (-5{{n}^{2}}p)=75{{m}^{3}}{{n}^{4}}{{p}^{2}}\]  

 

 

  Simplify \[3{{x}^{2}}{{y}^{2}}(5{{x}^{2}}-4xy+6{{y}^{2}})\]

(a) \[15{{x}^{4}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{4}}{{x}^{2}}\]       

(b) \[15{{x}^{2}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{2}}{{x}^{2}}\]

(c) \[15{{x}^{2}}{{y}^{3}}-12{{x}^{2}}{{y}^{2}}+6{{y}^{2}}{{y}^{2}}\]

(d) \[15x{{y}^{3}}-12{{x}^{3}}{{y}^{3}}+18{{x}^{2}}{{y}^{4}}\]

(e) None of these

 

Answer: (a)

Explanation:

\[3{{x}^{2}}{{y}^{2}}(-5{{x}^{2}}-4xy+6{{y}^{2}})=15{{x}^{4}}{{y}^{2}}-12{{x}^{3}}{{y}^{3}}+18{{y}^{4}}{{x}^{2}}\]    

 

 

  The product of \[(5{{x}^{2}}-6x+9)\] and \[(2x-3)\] is----.

(a) \[10{{x}^{3}}-27{{x}^{2}}+36x-27\]   

(b) \[10{{x}^{2}}-26{{x}^{2}}+36x-27\]

(c) \[10{{x}^{3}}-27{{x}^{2}}-36x-27\]     

(d) \[10{{x}^{3}}-27{{x}^{2}}+36x+27\]

(e) None of these  

 

Answer: (a)

Explanation:

\[(5{{x}^{2}}-6x+9)(2x-3)\] \[=10{{x}^{3}}-27{{x}^{2}}+36x-27\]  

 

 

  The product of \[(2{{x}^{2}}-5{{x}^{2}}-x+7)\] and \[(3-2x+{{x}^{2}})\] is -------.

(a) \[-3{{x}^{4}}+5{{x}^{3}}-17x+21\]     

(b) \[{{x}^{5}}+24{{x}^{4}}+5{{x}^{2}}+x+21\]

(c) \[8{{x}^{5}}+{{x}^{4}}-12+7x+1\]                       

(d) \[3{{x}^{5}}-4{{x}^{4}}+1{{x}^{3}}-5{{x}^{2}}-7x+2\]

(e) None of these  

 

Answer: (a)

Explanation:

\[=(2{{x}^{2}}-5{{x}^{2}}-x+7)(3-2x+{{x}^{2}})\] \[=6{{x}^{2}}-4{{x}^{3}}+2{{x}^{4}}-15{{x}^{2}}+10{{x}^{3}}-5{{x}^{4}}-3x\]\[2{{x}^{2}}-{{x}^{3}}+21-14x+7{{x}^{2}}\] \[=-3{{x}^{4}}+5{{x}^{3}}-17x+21\]  

 

 

  Find the quotient of polynomial if \[{{y}^{3}}-6{{y}^{2}}+9y-2\] is divided by \[y-2\].

(a) \[{{y}^{2}}+4y+1\]                                   

(b) \[{{y}^{2}}-4y+1\]

(c) \[{{y}^{2}}+4y-1\]                                     

(d) \[{{y}^{2}}-4y-1\]

(e) None of these  

 

Answer: (b)

Explanation:

\[y-2\overset{{{y}^{2}}-4y-1}{\overline{\left){\begin{align}   & {{y}^{3}}-6{{y}^{2}}+9y-2 \\  & \underline{\pm {{y}^{3}}\mp 2{{y}^{2}}} \\  & \,\,\,\,\,\,\,\,\,\,\,-4{{y}^{2}}+9y \\  & \,\,\,\,\,\,\,\,\,\,\,\underline{\mp 4{{y}^{2}}\pm 8y} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y-2 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm y\mp 2} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \\ \end{align}}\right.}}\]  

 

 

  Find the remainder when \[\text{5}{{\text{m}}^{\text{3}}}-\text{13}{{\text{m}}^{\text{2}}}+\text{15m}+\] 7 is divided by \[\text{4}-\text{3m}+{{\text{m}}^{\text{2}}}\].

(a) m-1                                

(b) m+1

(c) 2m+1                             

(d) 2m-1                              

(e) None of these  

 

Answer: (a)

Explanation:

\[\text{4}-\text{3m}+{{\text{m}}^{\text{2}}}\overset{\text{5m}+\text{2}}{\overline{\left){\begin{align}   & \text{5}{{\text{m}}^{\text{3}}}-\text{13}{{\text{m}}^{\text{2}}}+\text{15m}+\text{7} \\  & \underline{\pm \text{5}{{\text{m}}^{\text{3}}}\mp \text{l5}{{\text{m}}^{\text{2}}}\pm \text{2}0\text{m}} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2{{m}^{2}}-5m+7 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\mp 2{{m}^{2}}\mp 6m+8} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m-1 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm m\pm 1} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \\ \end{align}}\right.}}\]  

 

 

  Divide \[2{{x}^{2}}+3x+1\] by \[(x+1)\] then quotient is:

(a) \[2x+1\]                                                      

(b) \[5x+1\]

(c) \[4x+1\]                                                        

(d) \[6x+1\]

(e) None of these

 

Answer: (a)

Explanation:

\[x+1\overset{2x+1}{\overline{\left){\begin{align}   & 2{{x}^{2}}+3x+1 \\  & \underline{\pm 2{{x}^{2}}\pm 2x} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x+1 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\pm x\pm 1} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,00 \\ \end{align}}\right.}}\]

Quotient \[=2x+1\]

Remainder = 0  

 

 

  Divide \[(29x-6{{x}^{2}}-28)\] by \[(3x-4)\] and find the quotient

(a) \[2x+7\]                                       

(b) \[-2x+7\]

(c) \[2x-7\]                                         

(d) \[2x+6\]

(e) None of these

 

Answer: (b)

Explanation:

\[3x-4\overset{-2x+7}{\overline{\left){\begin{align}   & -6{{x}^{2}}+29x-28 \\  & \underline{\mp 6{{x}^{2}}\pm 8x} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,21x-28 \\  & \,\,\,\,\,\,\,\,\,\,\,\underline{\pm 21x\mp 28} \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,00 \\ \end{align}}\right.}}\]