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

Simplify Product Using Identity Simplify the Following Product \[ (2x^4 – 4x^2 + 1)(2x^4 – 4x^2 – 1) \] Solution: \[ = \left[(2x^4 – 4x^2)+1\right] \left[(2x^4 – 4x^2)-1\right] \] Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ = (2x^4 – 4x^2)^2 – 1^2 \] \[ = (2x^4)^2 + (4x^2)^2 – 2(2x^4)(4x^2) – 1 \] \[ = 4x^8

Simplify Product Using Identity Simplify the Following Product \[ (x^3 – 3x^2 – x)(x^2 – 3x + 1) \] Solution: \[ = x(x^2 – 3x – 1)(x^2 – 3x + 1) \] \[ = x\left[(x^2 – 3x)^2 – 1^2\right] \] Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ = x\left(x^4 + 9x^2 – 6x^3 – 1\right) \]

Simplify Product Using Identity Simplify the Following Product \[ (x^2 + x – 2)(x^2 – x + 2) \] Solution: \[ = \left[x^2 + (x – 2)\right] \left[x^2 – (x – 2)\right] \] Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ = (x^2)^2 – (x – 2)^2 \] \[ = x^4 – \left(x^2 – 4x + 4\right)

Simplify Product Using Identity Simplify the Following Product \[ \left(\frac{x}{2} – \frac{2}{5}\right) \left(\frac{2}{5} – \frac{x}{2}\right) – x^2 + 2x \] Solution: \[ = -\left(\frac{x}{2} – \frac{2}{5}\right)^2 – x^2 + 2x \] \[ = -\left[ \left(\frac{x}{2}\right)^2 -2\left(\frac{x}{2}\right)\left(\frac{2}{5}\right) +\left(\frac{2}{5}\right)^2 \right] – x^2 + 2x \] \[ = -\left( \frac{x^2}{4} -\frac{2x}{5} +\frac{4}{25} \right) – x^2 + 2x \]

Find the Value Using Identity Find the Value \[ x^2 + \frac{1}{x^2} = 79 \] Find: \[ x + \frac{1}{x} \] Solution: Using identity: \[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 79+2 \] \[ \left(x+\frac{1}{x}\right)^2 = 81 \] \[ x+\frac{1}{x} = \pm 9 \] Next Question / Full Exercise

Find the Value Using Identity Find the Value \[ x^2 + \frac{1}{x^2} = 66 \] Find: \[ x – \frac{1}{x} \] Solution: Using identity: \[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 66-2 \] \[ \left(x-\frac{1}{x}\right)^2 = 64 \] \[ x-\frac{1}{x} = \pm 8 \] Next Question / Full Exercise

Simplify Products Using Identity Simplify the Following Products \[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \] Solution: \[ \left(m + \frac{n}{7}\right)^3 \left(m – \frac{n}{7}\right) \] \[ = \left(m + \frac{n}{7}\right)^2 \left(m + \frac{n}{7}\right) \left(m – \frac{n}{7}\right) \] Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ = \left(m + \frac{n}{7}\right)^2 \left(m^2 – \frac{n^2}{49}\right) \] Again using identity: \[

Simplify Products Using Identity Simplify the Following Products \[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) \left(\frac{1}{4}a^2 + 9b^2\right) \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ \left(\frac{1}{2}a – 3b\right) \left(3b + \frac{1}{2}a\right) = \left(\frac{1}{2}a\right)^2 – (3b)^2 \] \[ = \frac{1}{4}a^2 – 9b^2 \] Now the expression becomes: \[ \left(\frac{1}{4}a^2 – 9b^2\right) \left(\frac{1}{4}a^2 + 9b^2\right) \]