Question

1. $x^{4}+64x=0$
2. $8x^{5}-125x^{2}=0$
3. $8x^{4}-x=0$
4. $-8x^{3}+125=0$

Explanation:

Problem 5: $x^4 + 64x = 0$

Step1: Factor out common term

$x(x^3 + 64) = 0$

Step2: Factor sum of cubes

$x(x + 4)(x^2 - 4x + 16) = 0$

Step3: Solve for each factor

$x=0$; $x+4=0 \implies x=-4$; solve $x^2-4x+16=0$:
$$x=\frac{4\pm\sqrt{16-64}}{2}=\frac{4\pm\sqrt{-48}}{2}=2\pm2i\sqrt{3}$$

Problem 6: $8x^5 - 125x^2 = 0$

Step1: Factor out common term

$x^2(8x^3 - 125) = 0$

Step2: Factor difference of cubes

$x^2(2x - 5)(4x^2 + 10x + 25) = 0$

Step3: Solve for each factor

$x^2=0 \implies x=0$ (double root); $2x-5=0 \implies x=\frac{5}{2}$; solve $4x^2+10x+25=0$:
$$x=\frac{-10\pm\sqrt{100-400}}{8}=\frac{-10\pm\sqrt{-300}}{8}=\frac{-5\pm5i\sqrt{3}}{4}$$

Problem 7: $8x^4 - x = 0$

Step1: Factor out common term

$x(8x^3 - 1) = 0$

Step2: Factor difference of cubes

$x(2x - 1)(4x^2 + 2x + 1) = 0$

Step3: Solve for each factor

$x=0$; $2x-1=0 \implies x=\frac{1}{2}$; solve $4x^2+2x+1=0$:
$$x=\frac{-2\pm\sqrt{4-16}}{8}=\frac{-2\pm\sqrt{-12}}{8}=\frac{-1\pm i\sqrt{3}}{4}$$

Problem 8: $-8x^3 + 125 = 0$

Step1: Rearrange into difference of cubes

$125 - 8x^3 = 0 \implies (5 - 2x)(25 + 10x + 4x^2) = 0$

Step2: Solve for each factor

$5-2x=0 \implies x=\frac{5}{2}$; solve $4x^2+10x+25=0$:
$$x=\frac{-10\pm\sqrt{100-400}}{8}=\frac{-10\pm\sqrt{-300}}{8}=\frac{-5\pm5i\sqrt{3}}{4}$$

Answer:

1. $x=0$, $x=-4$, $x=2+2i\sqrt{3}$, $x=2-2i\sqrt{3}$
2. $x=0$ (multiplicity 2), $x=\frac{5}{2}$, $x=\frac{-5+5i\sqrt{3}}{4}$, $x=\frac{-5-5i\sqrt{3}}{4}$
3. $x=0$, $x=\frac{1}{2}$, $x=\frac{-1+i\sqrt{3}}{4}$, $x=\frac{-1-i\sqrt{3}}{4}$
4. $x=\frac{5}{2}$, $x=\frac{-5+5i\sqrt{3}}{4}$, $x=\frac{-5-5i\sqrt{3}}{4}$