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Question

# Question 4 Expand each of the following, using suitable identities: (i) (x+2y+4z)2 (ii) (2x−y+z)2 (iii) (−2x+3y+2z)2 (iv) (3a−7b−c)2 (v) (−2x+5y−3z)2 (vi) [14a−12b+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) (2x−y+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+z2−4xy−2yz+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+4z2−12xy+12yz−8zx (iv) (3a−7b−c)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+c2−42ab+14bc−6ac (v) (−2x+5y−3z)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+9z2−20xy−30yz+12zx (vi)[14a−12b+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+1−14ab−b+12a=a216+b24+1−14ab−b+12a=a216+b24+1−ab4−b+a2

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