7th Class Mathematics Algebraic Expressions Algebraic identities

Algebraic identities

Category : 7th Class

*     Algebraic identities

 

Abatement of equality which holds, for all values of the variable is called algebraic identities.

Now we recall some important algebraic identities:

  • \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
  • \[{{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
  • \[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b)\]
  • \[{{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\]
  • \[{{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)\]or \[{{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)\]
  • \[{{(a-b)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab(a-b)\]
  • \[{{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)\]or \[(a+b)({{a}^{2}}-ab+{{b}^{2}})\]
  • \[{{a}^{3}}-{{b}^{3}}={{(a-b)}^{3}}+3ab(a-b)\]or \[(a-b)({{a}^{2}}+ab+{{b}^{2}})\]
  • \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=(a+b+c)\]\[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca)\]

               If \[(a+b+c)=0\]\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc\]  

 

 

Simplify:  \[{{(2p+3q-+4r)}^{2}}+{{(2p-3q-4r)}^{2}}\]

(a) \[2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\]

(b) \[-2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\]

(c) \[2(-4{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}+16rp)\]

(d)\[~2(5{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}-98rp)\]

(e) None of these  

 

Answer: (a)

Explanation   

Let us first solve,

\[{{\left[ 2p+3q+(-4r) \right]}^{2}}={{(2p)}^{2}}+{{(3q)}^{2}}\]\[+{{(-4r)}^{2}}+2(2p)(3q)+2(3q)(-4r)\] \[+2(-4r)(2p)=4{{p}^{2}}+4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}\]\[+12pq-24qr-16rp..........(i)\]

Now solve, \[~{{(2p-3q-4r)}^{2}}={{\left[ 2p+(-3q)+(-4r) \right]}^{2}}\]

\[={{(2p)}^{2}}+{{(-3q)}^{2}}+{{(-4r)}^{2}}+2(2p)(-3q)+2(-3q)\]\[(-4r)+2(-4r)(2p)\]

\[=4{{p}^{2}}+9{{q}^{2}}+16r2-12pq+24qr-16rp\text{ }.......\left( ii \right)\]

Adding, (i) & (ii) we get,

\[{{(2p+3a-4r)}^{2}}+{{(2p-3q-4r)}^{2}}\]

\[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp\]+ \[(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-12pq+24qr-16rp)\]                

\[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp+4{{p}^{2}}+9{{q}^{2}}\]    \[+16{{r}^{2}}-12pq+24qr-16rp\]                

\[=8{{p}^{2}}+18{{q}^{2}}+32{{r}^{2}}-32rp\]                

\[=2(4{{p}^{2}}+9{{q}^{2+}}16r2-16rp)\]  

 

 

 

 

  The expanded form of \[{{(2x+3y-5z)}^{2}}\] is:                

(a) \[4{{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}+12xy-30yz-20zx\]                

(b) \[5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36{{y}^{6}}+21x{{y}^{2}}\]                

(c) \[8{{a}^{2}}+{{a}^{3}}-{{c}^{2}}+7{{x}^{2}}+2{{c}^{2}}+9{{c}^{2}}y\]                

(d) \[1{{z}^{2}}-{{z}^{4}}-{{8}^{c}}-{{75}^{2}}+2{{c}^{2}}-98{{c}^{2}}\]                

(e) None of these                                

 

Answer: (a)                

Explanation

\[{{(2x+3y-5z)}^{2}}={{(2x)}^{2}}+{{(3y)}^{2}}+{{(-5z)}^{2}}\]\[+2(2x)(3y)+2(3y)(-5z)+2(5z)(2x)\text{ }\]                

               

 

 

Expand: \[{{(3a-b+4c)}^{2}}\]

(a) \[~5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy\]                

(b) \[9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac\]                

(c) \[2{{x}^{2}}+7{{y}^{2}}+2{{z}^{2}}+xy+3yz+2zx\]

(d) \[15a2+22{{b}^{2}}+2{{c}^{4}}+ab+4bc+102ca\]

(e) \[5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36y+21xy\]                                

 

Answer: (b)                

Explanation                

\[{{(3a-b+4c)}^{2}}={{(3a(-b)+4c)}^{2}}={{(3a)}^{2}}\]\[+{{(-b)}^{2}}+{{(4c)}^{2}}+2(3a)(-b)+2(-b)\] \[(4c)+2(4c)(3a)=9{{a}^{2}}+{{b}^{2}}+16{{c}^{2}}-6ab-8bc+24ac\]  

 

 

  Find the value of \[{{(3a+5b)}^{3}}.\]

(a) \[27{{a}^{3}}+125{{b}^{3}}+135{{a}^{2}}b+225a{{b}^{2}}\]

(b) \[27{{a}^{3}}-125{{b}^{3}}-135{{a}^{2}}b+2225a{{b}^{2}}\]

(c) \[27{{a}^{2}}+155{{b}^{2}}-135{{a}^{2}}b-225{{a}^{2}}b\]

(d) \[29{{a}^{2}}-156{{b}^{2}}-156{{a}^{2}}b-225{{a}^{4}}c\]

(e) None of these                                

 

Answer: (a)                

 

 

Find the Cube of  \[x-2y.\]

(a)\[~{{x}^{3}}+8{{y}^{2}}+6{{x}^{2}}y-12x{{y}^{3}}\]                     

(b) \[{{x}^{3}}-8{{y}^{3}}-6{{x}^{2}}y+12x{{y}^{2}}\]v                

(c)\[{{x}^{2}}+87y+7xy-7xy\]                     

(d) \[7x{{y}^{3}}-6x-74{{x}^{2}}y+2xy\]                

(e) None of these                  

 

Answer: (b)                

 

 

Evaluate: \[{{\left( -6p+\frac{1}{3}q-r \right)}^{2}}\]

(a) \[36{{p}^{2}}\frac{9}{1}{{p}^{2}}+{{5}^{2}}-4pq-\frac{2}{3}gr+12rp\]                

(b) \[36{{p}^{3}}\frac{9}{1}{{q}^{2}}+{{p}^{2}}+4qp-\frac{3}{2}qr+13rp\]

(c)\[~36{{p}^{2}}+\frac{1}{9}{{q}^{2}}+{{r}^{2}}-4qp-\frac{2}{3}qr+12rp\]                             

(d) \[36{{x}^{2}}-\frac{99}{8}{{s}^{2}}-{{s}^{2}}-4pq-\frac{2}{3}qr+14rp\]

(e) None of these  

 

Answer: (c)  

You need to login to perform this action.
You will be redirected in 3 sec spinner

Free
Videos