Question

I need help factor x^5 + y^5

Answer 1

`\boxed{\sf \textsf{Factor Is:}(x + y) * (x^4 - x^3 * y + x^2 * y^2 - x * y^3 + y^4)}`

Explanation:

To factor the expression
`\sf x^5 + y^5`, we can use the formula for the sum of fifth powers:

`\boxed{\sf a^5 + b^5 = (a + b) * (a^(5-1)b^0 - a^(5-2) * b^1 + a^(5-3) * b^2 - a^(5-4) * b^3 + a^(5-5)b^4)}`

In this case, a = x and b = y:

`\sf x^5 + y^5 = (x + y) * (x^4 - x^3 * y + x^2 * y^2 - x * y^3 + y^4)`

So, the factored form of x^5 + y^5 is:

`\sf (x + y) * (x^4 - x^3 y + x^2 y^2 - x y^3 + y^4)`

Answer 2

(x + y)(x⁴ - x³y + x²y² - xy³ + y⁴)

Explanation:

To

factor the expression

x⁵ + y⁵, we can use the

sum of fifth powers formula

. The sum of fifth powers formula is given by:

a⁵ + b⁵ = (a + b)(a⁴ - a³b + a²b² - ab³ + b⁴)

So, in our case, a=x and b=y, and the

factored expression

will be:

⇒ x⁵ + y⁵ =

(x + y)(x⁴ - x³y + x²y² - xy³ + y⁴)

Thus, we have factored x⁵ + y⁵.

`\hrulefill`

Additional Information:

Sum of Powers Formula (Generalized):

The sum of powers formula can be generalized to any positive integer

n

as follows:

aⁿ + bⁿ = (a + b)(aⁿ⁻¹ - aⁿ⁻²b + aⁿ⁻³b² - ... - abⁿ⁻² + bⁿ⁻¹)

Difference of Powers Formula (Generalized):

The difference of powers formula can be generalized to any positive integer

n

as follows:

aⁿ - bⁿ = (a - b)(aⁿ⁻¹ + aⁿ⁻²b + aⁿ⁻³b² + ... + abⁿ⁻² + bⁿ⁻¹)