Final answer:
The polynomial 9x⁴ - 49 is factored by recognizing it as a difference of squares. It factors to (3x² - 7)(3x² + 7), as both 9x⁴ and 49 are perfect squares. This results in the final factored form of the polynomial.
Step-by-step explanation:
To solve the polynomial 9x⁴ - 49 by factoring, we look for patterns or factorizations that can simplify the expression.
The given polynomial is a difference of squares because 9x⁴ is a perfect square (being (3x²)²) and 49 is also a perfect square (being 7²). The difference of squares factorization pattern is a² - b² = (a - b)(a + b), where a and b are any expressions for which a² and b² are defined.
For 9x⁴ - 49, we can assign a = 3x² and b = 7. The factorization is:
(3x² - 7)(3x² + 7)
The polynomial is now factored into a product of two binomials, which cannot be factored further using real numbers since neither of the resulting binomials is a perfect square. These are the final factors of the original polynomial.