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\( w^{4} - 625 = (w - 5)(w + 5)(w^{2} + 25) \)
Solución
We begin with the expression w⁴ - 625. Notice that 625 is equal to 25², so we can rewrite the expression as:
w⁴ - 625 = (w²)² - (25)²
This is a difference of two squares, which factors as:
(w² - 25)(w² + 25)
Now, look at the first factor, w² - 25. Since 25 is 5², we can factor it further as another difference of two squares:
w² - 25 = (w - 5)(w + 5)
Thus, the complete factorization of w⁴ - 625 is:
(w - 5)(w + 5)(w² + 25)
This is the factorization over the real numbers. (Note that w² + 25 can be factored further over the complex numbers, but typically the factorization given above is sufficient.)
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Simplifique esta solución Mind Expander
Did you know that \( w^{4} - 625 \) can be factored using the difference of squares? It turns out \( 625 \) can be rewritten as \( 25^2 \), so we can express it as \( (w^2)^2 - (25)^2 \). This gives us a neat factorization of \( (w^2 - 25)(w^2 + 25) \). Now, you can take it a step further by factoring \( w^2 - 25 \) again since it's another difference of squares! It becomes \( (w - 5)(w + 5) \). So, your complete factorization is \( (w - 5)(w + 5)(w^2 + 25) \). Remember, the term \( w^2 + 25 \) won't factor further over the reals, but it does provide complex solutions, which can be explored if you're feeling adventurous!
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