Educational

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ \left(\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab}\right)^2 \] \[ = \left(\frac{a}{bc}\right)^2 + \left(\frac{b}{ca}\right)^2 + \left(\frac{c}{ab}\right)^2 + 2\left(\frac{a}{bc}\right)\left(\frac{b}{ca}\right) + 2\left(\frac{b}{ca}\right)\left(\frac{c}{ab}\right) + 2\left(\frac{c}{ab}\right)\left(\frac{a}{bc}\right) \] \[ = \frac{a^2}{b^2c^2} + \frac{b^2}{c^2a^2} + \frac{c^2}{a^2b^2} + \frac{2}{c^2} […]

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \] \[ = \left(\frac{x}{y}\right)^2 + \left(\frac{y}{z}\right)^2 + \left(\frac{z}{x}\right)^2 + 2\left(\frac{x}{y}\right)\left(\frac{y}{z}\right) + 2\left(\frac{y}{z}\right)\left(\frac{z}{x}\right) + 2\left(\frac{z}{x}\right)\left(\frac{x}{y}\right) \] \[ = \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + \frac{2x}{z}

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (ab + bc + ca)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ (ab + bc + ca)^2 \] \[ = (ab)^2 + (bc)^2 + (ca)^2 + 2(ab)(bc) + 2(bc)(ca) + 2(ca)(ab) \] \[ = a^2b^2 + b^2c^2 + c^2a^2 + 2ab^2c

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (a^2 + b^2 + c^2)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ (a^2 + b^2 + c^2)^2 \] \[ = (a^2)^2 + (b^2)^2 + (c^2)^2 + 2(a^2)(b^2) + 2(b^2)(c^2) + 2(c^2)(a^2) \] \[ = a^4 + b^4 + c^4 + 2a^2b^2

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (2 + x – 2y)^2 \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (2 + x – 2y)^2 \] \[ = (2)^2 + x^2 + (2y)^2 + 2(2)(x) – 2(x)(2y) – 2(2)(2y) \] \[ = 4 + x^2 + 4y^2 + 4x

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (m + 2n – 5p)^2 \] Solution: Using identity: \[ (a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ca \] \[ (m + 2n – 5p)^2 \] \[ = m^2 + (2n)^2 + (5p)^2 + 2(m)(2n) – 2(2n)(5p) – 2(m)(5p) \] \[ = m^2 + 4n^2 + 25p^2 + 4mn

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (-3x + y + z)^2 \] Solution: Using identity: \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \] \[ (-3x + y + z)^2 \] \[ = (-3x)^2 + y^2 + z^2 + 2(-3x)(y) + 2(y)(z) + 2(z)(-3x) \] \[ = 9x^2 + y^2 + z^2 – 6xy

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (2a – 3b – c)^2 \] Solution: Using identity: \[ (x-y-z)^2 = x^2+y^2+z^2-2xy+2yz-2xz \] \[ (2a – 3b – c)^2 \] \[ = (2a)^2 + (3b)^2 + c^2 – 2(2a)(3b) + 2(3b)(c) – 2(2a)(c) \] \[ = 4a^2 + 9b^2 + c^2 – 12ab

Proof Using Algebraic Identity Prove that the Following Expression is Always Non-Negative \[ a^2 + b^2 + c^2 – ab – bc – ca \] Proof: \[ 2(a^2 + b^2 + c^2 – ab – bc – ca) \] \[ = a^2 + b^2 – 2ab + b^2 + c^2 – 2bc + c^2 +

Expansion Using Algebraic Identity Write the Following in Expanded Form \[ (a + 2b + c)^2 \] Solution: Using identity: \[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \] \[ (a + 2b + c)^2 \] \[ = a^2 + (2b)^2 + c^2 + 2(a)(2b) + 2(2b)(c) + 2(a)(c) \] \[ = a^2 + 4b^2 + c^2 + 4ab