To factor the polynomial [tex]\( x^8 - 625 \)[/tex] completely, let's follow these steps:
1. Recognize the given polynomial:
The polynomial we need to factor is [tex]\( x^8 - 625 \)[/tex].
2. Rewrite the constant term as a power:
Notice that 625 can be written as [tex]\( 25^2 \)[/tex]:
[tex]\[ 625 = 25^2 \][/tex]
3. Express the polynomial in a more recognizable form:
[tex]\[ x^8 - 625 = x^8 - (25)^2 \][/tex]
4. Identify this as a difference of squares:
Recall the difference of squares formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, [tex]\( a = x^4 \)[/tex] and [tex]\( b = 25 \)[/tex]:
[tex]\[ x^8 - 625 = (x^4)^2 - (25)^2 \][/tex]
Thus, we can apply the difference of squares formula:
[tex]\[ x^8 - 625 = (x^4 - 25)(x^4 + 25) \][/tex]
5. Factor each term further if possible:
Now, consider the factor [tex]\( x^4 - 25 \)[/tex]. It can also be expressed as a difference of squares since:
[tex]\[ 25 = (5)^2 \][/tex]
Hence,
[tex]\[ x^4 - 25 = (x^2)^2 - (5)^2 \][/tex]
Applying the difference of squares formula here again:
[tex]\[ x^4 - 25 = (x^2 - 5)(x^2 + 5) \][/tex]
6. Write the complete factorization:
Combining all parts, we get:
[tex]\[ x^8 - 625 = (x^2 - 5)(x^2 + 5)(x^4 + 25) \][/tex]
Therefore, the completely factored form of [tex]\( x^8 - 625 \)[/tex] is:
[tex]\[ (x^2 - 5)(x^2 + 5)(x^4 + 25) \][/tex]
