Cube of Binomial Expression Find the Cube of the Following Binomial Expression \[ \frac{1}{x}+\frac{y}{3} \] Solution: Using identity: \[ (a+b)^3 = a^3+b^3+3ab(a+b) \] \[ \left(\frac{1}{x}+\frac{y}{3}\right)^3 \] \[ = \left(\frac{1}{x}\right)^3 + \left(\frac{y}{3}\right)^3 + 3\left(\frac{1}{x}\right)\left(\frac{y}{3}\right) \left(\frac{1}{x}+\frac{y}{3}\right) \] \[ = \frac{1}{x^3} + \frac{y^3}{27} + \frac{y}{x} \left(\frac{1}{x}+\frac{y}{3}\right) \] \[ = \frac{1}{x^3} + \frac{y^3}{27} + \frac{y}{x^2} + \frac{y^2}{3x} \] Next […]

Simplify Expression Using Identity Simplify the Following Expression \[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (x+y-2z)^2 = x^2+y^2+(2z)^2+2xy-2(y)(2z)-2(x)(2z) \] \[ = x^2+y^2+4z^2+2xy-4yz-4xz \] \[ (x+y-2z)^2-x^2-y^2-3z^2+4xy \] \[ = x^2+y^2+4z^2+2xy-4yz-4xz -x^2-y^2-3z^2+4xy \] \[ = z^2+6xy-4yz-4xz \] Next Question / Full Exercise

Simplify Expression Using Identity Simplify the Following Expression \[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \] Solution: Using identity: \[ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca \] \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ \left(x+\frac{y}{2}+\frac{z}{3}\right)^2 = x^2+\frac{y^2}{4}+\frac{z^2}{9} +xy+\frac{yz}{3}+\frac{2xz}{3} \] \[ \left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 = \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16} +\frac{xy}{3}+\frac{yz}{6}+\frac{xz}{4} \] \[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \] \[ = \frac{7x^2}{4} +\frac{41y^2}{36} +\frac{127z^2}{144} +\frac{8xy}{3} +\frac{13yz}{6} +\frac{29xz}{12} \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (x^2+y^2-z^2)^2-(x^2-y^2+z^2)^2 \] Solution: Using identity: \[ a^2-b^2=(a-b)(a+b) \] \[ = \left[(x^2+y^2-z^2)-(x^2-y^2+z^2)\right] \left[(x^2+y^2-z^2)+(x^2-y^2+z^2)\right] \] \[ = (2y^2-2z^2)(2x^2) \] \[ = 4x^2(y^2-z^2) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (2x+p-c)^2-(2x-p+c)^2 \] Solution: Using identity: \[ a^2-b^2=(a-b)(a+b) \] \[ = \left[(2x+p-c)-(2x-p+c)\right] \left[(2x+p-c)+(2x-p+c)\right] \] \[ = (2p-2c)(4x) \] \[ = 8x(p-c) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 + (a-b+c)^2 + (a+b-c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (a+b+c)^2 + (a-b+c)^2 + (a+b-c)^2 \] \[ = 3a^2+3b^2+3c^2+2ab-2bc+2ca \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 – (a-b+c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b+c)^2 – (a-b+c)^2 \] \[ = a^2+b^2+c^2+2ab+2bc+2ca \] \[ – (a^2+b^2+c^2-2ab-2bc+2ca) \] \[ = 4ab+4bc \] \[ = 4b(a+c) \] Next Question / Full Exercise

Simplify Using Algebraic Identity Simplify \[ (a+b+c)^2 + (a-b+c)^2 \] Solution: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (a-b+c)^2 = a^2+b^2+c^2-2ab-2bc+2ca \] \[ (a+b+c)^2 + (a-b+c)^2 \] \[ = a^2+b^2+c^2+2ab+2bc+2ca \] \[ + a^2+b^2+c^2-2ab-2bc+2ca \] \[ = 2a^2+2b^2+2c^2+4ca \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ 4x^2 + y^2 + 25z^2 + 4xy – 10yz – 20zx \] Given: \[ x=4,\quad y=3,\quad z=2 \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ 4x^2 + y^2 + 25z^2 + 4xy – 10yz – 20zx \] \[ = (2x)^2 + y^2 + (5z)^2

Find the Value Using Identity Find the Value \[ a+b+c=9 \] \[ ab+bc+ca=23 \] Find: \[ a^2+b^2+c^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (9)^2 = a^2+b^2+c^2+2(23) \] \[ 81 = a^2+b^2+c^2+46 \] \[ a^2+b^2+c^2 = 81-46 \] \[ =35 \] Next Question / Full Exercise