7th Class Mathematics Algebraic Expressions Algebraic identities

     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)