Question

Factorise the following:

64x⁴ – 1000x

Answer 1

Given expression: 64x⁴ – 1000x

Step-wise calculation:

1. Take out the common factor (4x) the greatest common factor of 64 and 1000 includes 4, and the lowest power of (x) in terms is (x):

64x⁴ – 1000x = 4x(16x³ – 250)

2. Now we have (16x³ – 250). 

Notice both terms are cubes:

16x³ = (2x)³, `250 = (5 xx sqrt(3)(2))^3` actually, 250 is not a perfect cube, so it may be better to rewrite the expression differently.

Actually, 250 can be written as 125 × 2 = 5³ × 2, which is not a perfect cube.

So instead, reconsider the approach to factor 16x³ – 250.

Observe: 16x³ – 250 can be factored as a difference of cubes if rewritten as 64x³ – 1000.

But we have 16x³ – 250, not cubed terms exactly.

3. Let’s go back and see if the initial common factor could be larger:

The original terms are (64x⁴) and (1000x).

The GCF of (64) and (1000) is 4.

The lowest power of (x) is (x).

So factor out (4x): 4x(16x³ – 250).

4. 16x³ – 250 can be factored by recognizing as difference of cubes if expressed as:

`16x^3 - 250 = (2x)^3 – (5sqrt(3)(2))^3`

Since 250 is not a perfect cube, it is better to factor as difference of squares or by further factorization.

5. Alternatively, factor (16x³ – 250) as 16x³ – 250 = 2(8x³ – 125).

Here, 8x³ = (2x)³ and 125 = 5³, so this becomes 2((2x)³ – 5³).

6. Use the difference of cubes factorization formula: 

a³ – b³ = (a – b)(a² + ab + b²) 

Applying for a = 2x, b = 5: 

(2x – 5)((2x)² + (2x)(5) + 5²)

= (2x – 5)(4x² + 10x + 25)

7. So, 16x³ – 250 = 2(2x – 5)(4x² + 10x + 25).

8. Putting everything together: 

64x⁴ – 1000x = 4x × 2 × (2x – 5)(4x² + 10x + 25)

= 8x(2x – 5)(4x² + 10x + 25)