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Lena [83]
2 years ago
8

Someone please help me

Mathematics
1 answer:
Aleonysh [2.5K]2 years ago
7 0

Step-by-step 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:

<h3>Difference of Two Squares</h3>

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

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

<h3>Sum and Difference of Two Cubes</h3>

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 <em>u </em>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)

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