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Answer :
To factor the expression [tex]\(64x^4 + 625\)[/tex] completely, we can recognize that this is a sum of squares. Let's break it down step by step:
1. Identify the Perfect Squares:
- Notice that [tex]\(64\)[/tex] and [tex]\(625\)[/tex] are both perfect squares.
- [tex]\(64 = 8^2\)[/tex]
- [tex]\(625 = 25^2\)[/tex]
2. Rewriting the Expression:
- The expression [tex]\(64x^4 + 625\)[/tex] can be rewritten as [tex]\((8x^2)^2 + (25)^2\)[/tex].
- This represents a sum of squares: [tex]\((8x^2)^2 + (25)^2\)[/tex].
3. Factoring Using Complex Numbers:
- A sum of squares can be factored using complex numbers. The expression [tex]\((a^2 + b^2)\)[/tex] can be factored into [tex]\((a + bi)(a - bi)\)[/tex].
- Here, [tex]\(a = 8x^2\)[/tex] and [tex]\(b = 25\)[/tex].
- So, [tex]\((8x^2)^2 + (25)^2\)[/tex] factors to [tex]\((8x^2 + 25i)(8x^2 - 25i)\)[/tex].
Therefore, the factored form of [tex]\(64x^4 + 625\)[/tex] over the complex numbers is [tex]\((8x^2 + 25i)(8x^2 - 25i)\)[/tex].
1. Identify the Perfect Squares:
- Notice that [tex]\(64\)[/tex] and [tex]\(625\)[/tex] are both perfect squares.
- [tex]\(64 = 8^2\)[/tex]
- [tex]\(625 = 25^2\)[/tex]
2. Rewriting the Expression:
- The expression [tex]\(64x^4 + 625\)[/tex] can be rewritten as [tex]\((8x^2)^2 + (25)^2\)[/tex].
- This represents a sum of squares: [tex]\((8x^2)^2 + (25)^2\)[/tex].
3. Factoring Using Complex Numbers:
- A sum of squares can be factored using complex numbers. The expression [tex]\((a^2 + b^2)\)[/tex] can be factored into [tex]\((a + bi)(a - bi)\)[/tex].
- Here, [tex]\(a = 8x^2\)[/tex] and [tex]\(b = 25\)[/tex].
- So, [tex]\((8x^2)^2 + (25)^2\)[/tex] factors to [tex]\((8x^2 + 25i)(8x^2 - 25i)\)[/tex].
Therefore, the factored form of [tex]\(64x^4 + 625\)[/tex] over the complex numbers is [tex]\((8x^2 + 25i)(8x^2 - 25i)\)[/tex].
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