1. **State the problems:** (a) Factorize the expression $$a^4 - 6a^2 + 1$$. (b) Find the value of $$8x^3 + \frac{1}{27x^3}$$. (c) Find the value of $$3x^5 - \frac{1}{243x^5}$$. --- 2. **Part (a): Factorize $$a^4 - 6a^2 + 1$$** - Recognize it as a quadratic in terms of $$a^2$$: $$ (a^2)^2 - 6(a^2) + 1 $$. - Use the identity for factoring expressions of the form $$x^2 - 2xy + y^2 - z$$: $$a^4 - 6a^2 + 1 = (a^2)^2 - 2 \cdot a^2 \cdot 3 + 3^2 - 8$$ (rewriting to complete the square is tricky here, so instead use the difference of squares approach). - Rewrite as: $$a^4 - 6a^2 + 1 = (a^2)^2 - 2 \cdot a^2 \cdot 1 + 1^2 - 4a^2$$ - This can be expressed as: $$ (a^2 - 1)^2 - (2a)^2 $$ - Apply difference of squares formula: $$ (a^2 - 1 + 2a)(a^2 - 1 - 2a) $$ - So the factorization is: $$ (a^2 + 2a - 1)(a^2 - 2a - 1) $$ --- 3. **Part (b): Find $$8x^3 + \frac{1}{27x^3}$$** - Notice that $$8x^3 = (2x)^3$$ and $$\frac{1}{27x^3} = \left(\frac{1}{3x}\right)^3$$. - Let $$y = 2x + \frac{1}{3x}$$. - Then, by the identity for cubes: $$y^3 = (2x)^3 + 3(2x)^2 \left(\frac{1}{3x}\right) + 3(2x) \left(\frac{1}{3x}\right)^2 + \left(\frac{1}{3x}\right)^3$$ - Simplify the middle terms: $$y^3 = 8x^3 + 3 \cdot 4x^2 \cdot \frac{1}{3x} + 3 \cdot 2x \cdot \frac{1}{9x^2} + \frac{1}{27x^3}$$ $$= 8x^3 + 4x + \frac{2}{3x} + \frac{1}{27x^3}$$ - Since the problem only asks for $$8x^3 + \frac{1}{27x^3}$$, rearrange: $$8x^3 + \frac{1}{27x^3} = y^3 - 3y$$ - Without a specific value for $$x$$, this is the simplified form. --- 4. **Part (c): Find $$3x^5 - \frac{1}{243x^5}$$** - Recognize $$3x^5 = 3x^5$$ and $$\frac{1}{243x^5} = \frac{1}{3^5 x^5} = \left(\frac{1}{3x}\right)^5$$. - Let $$z = 3x - \frac{1}{3x}$$. - Use the binomial expansion for $$z^5$$: $$z^5 = (3x)^5 - 5(3x)^4 \left(\frac{1}{3x}\right) + 10(3x)^3 \left(\frac{1}{3x}\right)^2 - 10(3x)^2 \left(\frac{1}{3x}\right)^3 + 5(3x) \left(\frac{1}{3x}\right)^4 - \left(\frac{1}{3x}\right)^5$$ - Simplify terms and isolate $$3x^5 - \frac{1}{243x^5}$$: $$3x^5 - \frac{1}{243x^5} = z^5 + 5z^3 + 10z$$ - Without a specific $$x$$ value, this is the simplified expression. --- **Final answers:** (a) $$ (a^2 + 2a - 1)(a^2 - 2a - 1) $$ (b) $$ 8x^3 + \frac{1}{27x^3} = \left(2x + \frac{1}{3x}\right)^3 - 3 \left(2x + \frac{1}{3x}\right) $$ (c) $$ 3x^5 - \frac{1}{243x^5} = \left(3x - \frac{1}{3x}\right)^5 + 5 \left(3x - \frac{1}{3x}\right)^3 + 10 \left(3x - \frac{1}{3x}\right) $$