Certainly! Let's factor the polynomial [tex]\( x^6 - 25 \)[/tex] step-by-step:
1. Identify the polynomial: We start with the polynomial [tex]\( x^6 - 25 \)[/tex].
2. Recognize the difference of squares: Notice that [tex]\( x^6 - 25 \)[/tex] can be seen as a difference of squares. Recall that [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]. In our case:
[tex]\[
x^6 - 25 = (x^3)^2 - 5^2
\][/tex]
Here, [tex]\( a = x^3 \)[/tex] and [tex]\( b = 5 \)[/tex].
3. Apply the difference of squares formula: Using the formula [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex] with [tex]\( a = x^3 \)[/tex] and [tex]\( b = 5 \)[/tex]:
[tex]\[
(x^3)^2 - 5^2 = (x^3 - 5)(x^3 + 5)
\][/tex]
4. Result: Therefore, the polynomial [tex]\( x^6 - 25 \)[/tex] factors as:
[tex]\[
x^6 - 25 = (x^3 - 5)(x^3 + 5)
\][/tex]
So, the factorization of the polynomial [tex]\( x^6 - 25 \)[/tex] is:
[tex]\[
(x^3 - 5)(x^3 + 5)
\][/tex]
This gives us the factored form of the initial polynomial.
