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Question

Question 4
Expand each of the following, using suitable identities:
(i) (x+2y+4z)2
(ii) (2xy+z)2
(iii) (2x+3y+2z)2
(iv) (3a7bc)2
(v) (2x+5y3z)2
(vi) [14a12b+1]2

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Solution

(i) (x+2y+4z)2 [Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca]=x2+(2y)2+(4z)2+2x×2y+2×2y×4z+2×4z×xx2+4y2+16z2+4xy+16yz+8zx

(ii) (2xy+z)2 [Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca]=(2x)2+(y)2+(z)2+2×(2x)(y)+2(y)(z)+2×2xz=4x2+y2+z24xy2yz+4zx

(iii) (2x+3y+2z)2 [Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca]=(2x)2+(3y)2+(2z)2+2(2x)(3y)+2(3y)(2z)+2(2z)(2x)=4x2+9y2+4z212xy+12yz8zx

(iv) (3a7bc)2 [Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca]=(3a)2+(7b)2+(c)2+2(3a)(7b)+2(7b)(c)+2(c)(3a)=9a2+49b2+c242ab+14bc6ac

(v) (2x+5y3z)2 [Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca]=(2x)2+(5y)2+(3z)2+2(2x)(5y)+2(5y)(3z)+2(3z)(2x)=4x2+25y2+9z220xy30yz+12zx

(vi)[14a12b+1]2[Using the identity(a+b+c)2=a2+b2+c2+2ab+2bc+2ca]

(14a)2+(12b)2+(1)2+2(14a)(12b)+2(12b)×1+2(1)×14a=116a2+14b2+114abb+12a=a216+b24+114abb+12a=a216+b24+1ab4b+a2

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