factor these expressions

A) x^4+x^2+1
B) x^10+x^5+1

A) (x^2 + x + 1) (x^2 - x +1)

B) (x^2 + x + 1) (x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 )

just curious, how did you get the second one??

thanks!

To factor the given expressions, we'll use a technique called factoring by grouping. Let's break down each expression one by one:

A) x^4 + x^2 + 1

To factor this expression, we'll look for common factors among the terms. Unfortunately, there are no common factors in this case. However, notice that this expression has a similar form to a quadratic trinomial.

Let's introduce a substitution: Let's say y = x^2. By substituting y for x^2, our expression becomes:
y^2 + y + 1

Now, this expression can be factored further using the quadratic formula or by completing the square. However, neither of those methods will result in factoring the expression into linear factors with real coefficients.

Hence, the factored form of expression A) x^4 + x^2 + 1 is (x^2 + (sqrt(3)i)x + 1)(x^2 - (sqrt(3)i)x + 1), where i is the imaginary unit.

B) x^10 + x^5 + 1

Similarly, we'll attempt to factor this expression. Notice that this expression has a similar form to a quadratic trinomial as well. Let's introduce a substitution here too: Let's say y = x^5. By substituting y for x^5, our expression becomes:
y^2 + y + 1

Now, we are left with a quadratic trinomial. We can factor it using any of the available methods, such as completing the square or quadratic formula. However, just like in expression A), this quadratic trinomial cannot be factored further into linear factors with real coefficients.

Therefore, the factored form of expression B) x^10 + x^5 + 1 is (x^5 + (sqrt(3)i)(x^(5/2)) + 1)(x^5 - (sqrt(3)i)(x^(5/2)) + 1), where i is the imaginary unit.

In both cases, the expressions cannot be factored into linear factors with real coefficients. Thus, the substitutions do not help factor the expressions further.