6th Class Mathematics Algebra Operations on Algebraic Expressions

  Operations on Algebraic Expressions

 

 

When constant and variables are linked with any of the following fundamental arithmetic operations, addition, subtraction, multiplication and division. The solution of the expression is obtained by simplification of the expression.  

 

 

 

Simplify,\[\text{2}{{\text{X}}^{\text{2}}}\text{-5}{{\text{X}}^{\text{2}}}\text{+6}\]?

(a) \[\text{2}{{\text{X}}^{\text{2}}}-\text{5}{{\text{X}}^{\text{2}}}\]                      

(b) \[\text{5}{{\text{X}}^{\text{2}}}\text{+6}\]

(c) \[-3{{x}^{2}}+6\]       

(d) All of these

(e) None of these

 

Answer: (c)                

Explanation

\[2{{X}^{2}}-5{{X}^{2}}+6=-3{{X}^{2}}+6.\]  

 

 

Addition of Algebraic Expression

Addition is possible even if terms are like. The addition of two unlike terms is possible and their addition is in the same form. Addition of \[2x+3x=5x\] but the addition of \[2x+3y=2x+3y.\]  

 

 

Add the following polynomials,\[~{{\text{X}}^{\text{3}}}-\text{3}{{\text{X}}^{\text{2}}}-\text{6X+10}\]and \[\text{4}{{\text{X}}^{\text{3}}}\text{+10}{{\text{X}}^{\text{2}}}\text{+15X}-20\]?

(a) \[\text{5}{{\text{X}}^{\text{3}}}\text{+7}{{\text{X}}^{\text{2}}}\text{+9X}-10\]               

(b) \[\text{5}{{\text{X}}^{\text{2}}}\text{+6+45}\]

(c) \[\text{5}{{\text{X}}^{\text{2}}}\text{+3}{{\text{X}}^{2}}\text{+6}\]                

(d) All of these

(e) None of these  

 

Answer: (a)

Explanation 

\[\begin{align}   & {{\text{X}}^{\text{3}}}-\text{3}{{\text{X}}^{\text{2}}}-\text{6X+10} \\  & \frac{\text{4}{{\text{X}}^{\text{3}}}\text{+10}{{\text{X}}^{\text{2}}}\text{+15X}-20}{\text{5}{{\text{X}}^{\text{3}}}\text{+7}{{\text{X}}^{\text{2}}}\text{+9X}-10} \\ \end{align}\]   Alternative Method \[({{X}^{3}}-3{{X}^{2}}-6X+10+\left( 4{{X}^{3}}+10{{X}^{2}}+15X-20 \right)\]\[=X3-3{{X}^{2}}-6X+10+4{{X}^{3}}+10{{X}^{2}}+15X-20\] \[={{X}^{3}}+4{{X}^{3}}-3{{X}^{2}}+10{{X}^{2}}-6X+15X+10-20\]\[=5{{X}^{3}}+7{{X}^{2}}+9X-10\]  

 

 

Subtraction of Algebraic Expression

Subtraction of two like terms is same as the subtraction of 2 mangoes from 4mangoes. Number of mangoes are constant and the name, mangoes are like terms for both the numbers 2 and 4. The subtraction of 2 bananas from 4 mangoes is not possible.                                 

 

 

Subtract: \[4{{X}^{2}}Y-3XY+5X\]from \[10{{X}^{2}}6XY+15X-25\] ?

(a) \[8{{x}^{3}}+2{{x}^{2}}+9x\]                 

(b) \[6{{x}^{2}}y-3xy+10x-25\]

(c) \[~5x-3{{X}^{2}}+6\]

(d) All of these                

(e) None of these                

 

Answer: (b)                

Explanation

\[=~(10{{X}^{2}}Y-6XY+15X-25)-(4{{X}^{2}}Y-3XY+5X)\]                

\[=~10{{X}^{2}}Y-6XY+15X-25-4{{X}^{2}}Y+3XY-5X\] \[=10{{X}^{2}}4{{X}^{2}}Y-6XY+3XY+15X-5X-25\]                

\[6{{X}^{2}}Y-3XY+10X-25\]  

 

 

Multiplication of Algebraic Expression

The following steps are used to perform the multiplication of algebraic expression.

Ist : Write the sign of the resulting product according to the following rules,                

\[\left( \frac{+x+=+,+x-=-}{-x-=+,-x+=-} \right)\]

IInd : Write the product of constant.

IIIrd : Write the product of variable according to the following rule, \[({{a}^{m}}\times {{a}^{n}}={{a}^{m+n}})\]  

 

 

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

(a) \[{{a}^{4}}{{a}^{2}}{{b}^{2}}-2a{{b}^{3}}-{{b}^{4}}\]                 

(b) \[6{{a}^{2}}b-3ab+10a-25\]

(c) \[5a-3{{a}^{2}}+6~\] 

(d) All of these

(e) None of these  

 

Answer: (a)                

Explanation                

\[({{a}^{2}}+ab+{{b}^{2}})({{a}^{2}}-ab-{{b}^{2}})\] \[={{a}^{2}}({{a}^{2}}-ab-{{b}^{2}})+ab({{a}^{2}}-ab-{{b}^{2}})+{{b}^{2}}({{a}^{2}}-ab-{{b}^{2}})\] \[={{a}^{4}}-{{a}^{3}}b-{{a}^{2}}{{b}^{2}}+{{a}^{3}}b-{{a}^{2}}{{b}^{2}}-a{{b}^{3}}+{{a}^{2}}{{b}^{2}}-a{{b}^{3}}-{{b}^{4}}\] \[={{a}^{4}}-{{a}^{2}}{{b}^{2}}-a{{b}^{3}}-a{{b}^{3}}-{{b}^{4}}\]                 \[={{a}^{4}}-{{a}^{4}}{{b}^{2}}-2a{{b}^{3}}-{{b}^{4}}\]  

 

 

  Division of Algebraic Expression

The following steps are used to perform the division of the algebraic expression

Ist : First keep the polynomials which is to be divided in division form.

IInd: Divide first term of dividend by 1st term of divisor and write quotient.

IIIrd: Write the product of quotient x divisor, below the dividend and subtract it from dividend.

IV th: Repeat this process until the degree of remainder becomes less than divisor  

 

 

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

(a) \[~3x+2\]                      

(b) \[~2x+1\]

(c) \[5x-3\]                          

(d) All of these

(e) None of these  

 

Answer: (b)  

Explanation

\[x+1\overset{2x+1}{\overline{\left){\begin{align}   & 2{{x}^{2}}+3x+1 \\  & \frac{\pm 2{{x}^{2}}\pm 2x}{\begin{align}   & x+1 \\  & \frac{\pm x\pm 1}{0} \\ \end{align}} \\ \end{align}}\right.}}\] Quotient of the division \[=2x+1\] is the solution of the expression.  

 

 

 

 

Add the following expression, \[6{{x}^{2}}-3by+4cz,7by-8ax-5cz\]and  \[9cz2by+2ax\] ?

(a)\[{{x}^{2}}-2by+8cz-6ax\]                      

(b) \[6{{x}^{2}}+2by+8cz-6ax\]

(c) \[6{{x}^{2}}+2by+8cz-6ax\]                   

(d) All of these

(e) None of these  

 

Answer: (c)

Explanation                

\[(6{{x}^{2}}-3by+4cz)+(7by-8ax-5cz)+(9cz-2by+2ax)\] \[=6{{x}^{2}}-3by+4cz+7by-8ax-5cz+9cz-2by+2ax\] \[=6{{x}^{2}}+2by+8cz-6ax\]  

 

 

Subtract the following, \[{{x}^{3}}-3{{x}^{2}}-5x+4\]from\[3{{x}^{3}}-{{x}^{2}}2x-4\]?

(a) \[2{{x}^{3}}+2{{x}^{2}}+7x-8~~~\]     

(b) \[-2{{x}^{5}}-7x+8\]

(c) \[2{{x}^{3}}-2{{x}^{2}}-4x-8\]               

(d) All of these

(e) None of these  

 

Answer: (a)

Explanation

\[3{{x}^{3}}-2+2x-4-({{x}^{3}}-3{{x}^{2}}-5x+4)\] \[=3{{x}^{3}}-{{x}^{2}}+2x-4-{{x}^{3}}+3{{x}^{2}}+5x-4\] \[2{{x}^{3}}+2{{x}^{2}}+7x-8\]  

 

 

A Find the product of \[(2x+4y)(3x-2y).\]

(a) \[6{{x}^{2}}+8xy-8y2\]            

(b) \[~{{x}^{2}}-8xy-8{{y}^{2}}\]

(c) \[8{{x}^{2}}-xy-8{{y}^{2}}\]   

(d) All of these

(e) None of these  

 

Answer: (a)  

Explanation

\[(2x+4y)(3x-2y)=6{{x}^{2}}-4xy+12xy-8{{y}^{2}}=6{{x}^{2}}+8xy-8{{y}^{2}}\]  

 

 

Simplify the following, \[{{x}^{o}}-xy-8{{y}^{o}}.\]

(a) \[(xy+7)\]                     

(b) \[(xy-7)\]

(c) \[-(xy+7)\]    

(d) All of these

(e) None of these  

 

Answer: (c)

Explanation

\[{{x}^{o}}-xy-8{{y}^{o}}=1-xy-8=-xy-7=-(xy+7)\]  

 

 

Simplify the following, \[6{{x}^{2}}-4x\div \text{ }x.\]

(a) \[6x-4\]                         

(b) \[6x+4\]

(c) \[4+6x\]                         

(d) All of these                 

(e) None of these  

 

Answer: (a)                

Explanation                

\[\overset{6x-4}{\overline{\left){\begin{align}   & 6{{x}^{2}}-4x \\  & \frac{\mp 6{{x}^{2}}}{\frac{\begin{align}   & -4x \\  & \mp 4x \\ \end{align}}{00}=(6x-4)} \\ \end{align}}\right.}}\]  

 

 

The product of 2 with an unknown number \[x\]is expressed by \[2x.\] Find the constant of the resulting product.

(a) 1                                                      

(b) \[x\]

(c) 2                                                      

(d) All of these                 

(e) None of these  

 

Answer: (c)                

Explanation

Numbers are the constant term.  

 

 

The alternate name of variable of a term other than variable is?

(a) Literal                                            

(b) Constant

(c) Coefficient                                  

(d) All of these                 

(e) None of these  

 

Answer: (a)

Explanation

Variables are also known as literal.  

 

 

If a term has one constant and one variable. They are linked to each other by which one of the following arithmetic operations if there is no sign of any operations between the term?

(a) Multiplication                             

(b) Addition

(c) Subtraction                                 

(d) All of these                 

(e) None of these  

 

Answer: (a)

Explanation

No sign between the term represent the product sign.        

 

 

Consider the following two statements.

Statement 1 : The division of two unlike terms, without constant can not be reduced to its lowest term.

Statement 2 : The product of two unlike terms is square of each of the term.

(a) Statement 1 and 2 are true  

(b) Statement 1 is true and 2 is false

(c) Statement 1 is false and 2 is true

(d) All of these                                                 

(e) None of these  

 

Answer: (b)                

Explanation

Let two unlike terms without constant term are x and y. The division of the terms \[=\frac{x}{y}\] and it cannot be reduced.    

 

 

Find the coefficient of 4 in the term, \[6abc+4xyz\]?

(a) \[x\]                

(b) \[y\]

(c) \[z\]                

(d) All of these

(e) None of these  

 

Answer: (d)

Explanation

The coefficient of 4 in the term \[6abc+4xyz\] is\[~xyz.\]      

 

 

• There can be various values for a variable in an expression.
• Constant has a fixed value.
• Variables are also known as literals.
• Elementary algebra deals with the properties of operations on real numbers.
• The geometrical study of algebra is known as the algebraic geometry.  

 

 

• Numbers are the constant terms in the algebraic expressions.
• Letters are the variable terms in the algebraic expressions.
• In the algebraic expression variables are the coefficient of numericals and numericals are the coefficient of variables.
• The greatest power of all the terms in an algebraic expression is the degree of the expression. In the term, variables and constant are linked to each other by multiplication.