Question

Someone please help me

Answer 1

Explanation:

Part A:
`u^6` can be written as the square of u³, or
`(u^3)^2`. Similarly,
`v^6=(v^3)^2`. Hence, we can write this as a difference of two squares by writing it as

`(u^3)^2-(v^3)^2`

Part B:

Difference of Two Squares

We can first factor a difference of two squares a² - b² into (a+b)(a-b). Here, a would be u³ and b would be v³.

`(u^3+v^3)(u^3-v^3)`

Sum and Difference of Two Cubes

We can factor this further by the use of two special formulas to factor a sum of two cubes and a difference of two cubes. These formulas are as follows:

`a^3+b^3=(a+b)(a^2-ab+b^2)\\a^3-b^3=(a-b)(a^2+ab+b^2)`

Since u³ + v³ is a sum of two cubes, let's rewrite it.

`u^3+v^3=(u+v)(u^2-uv+v^2)`

Since u³ - v³ is a difference of two cubes, we can rewrite it as well.

`u^3-v^3=(u-v)(u^2+uv+v^2)`

Now, let's multiply them together again to get the final factored form.

`u^6-v^6=(u+v)(u^2-uv+v^2)(u-v)(u^2+uv+v^2)`

Part C:

If we want to factor
`x^6-1` completely, we can just see that x to the sixth power is just
`x^6` and 1 to the sixth power is just 1. Hence, x can substitute for u and 1 can substitute for v.

`x^6-1=(x+1)(x^2-x(1)+1^2)(x-1)(x^2+x(1)+1^2)\\x^6-1=(x+1)(x^2-x+1)(x-1)(x^2+x+1)`

We can repeat this for
`x^6-64`, as 64 is just 2 to the sixth power.

`x^6-64=(x+2)(x^2-x(2)+2^2)(x-2)(x^2+x(2)+2^2)\\x^6-64=(x+2)(x^2-2x+4)(x-2)(x^2+2x+4)`