Question
Pagrowsume 1) \( a^{2}-9 \) 2) \( b^{2}+10 b+25 \) 3) \( 25 x^{2}-16 \) 4) \( 9 x^{2}-12 x y+4 y^{2} \) 5) \( 5 a^{2}-20 a b \) 6) \( 7 x^{3}-14 x^{5} \) 7) \( 3 a-3 b+a x-b x \) 8) \( 17 x^{2}-17 \) 9) \( 8 a^{2} b^{2}-72 a^{2} c^{2} \)
Ask by Delgado Mathis. in Russia
Jan 24,2025
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Answer
Разложение на множители:
1) \( a^{2} - 9 = (a - 3)(a + 3) \)
2) \( b^{2} + 10b + 25 = (b + 5)^{2} \)
3) \( 25x^{2} - 16 = (5x - 4)(5x + 4) \)
4) \( 9x^{2} - 12xy + 4y^{2} = (3x - 2y)^{2} \)
5) \( 5a^{2} - 20ab = 5a(a - 4b) \)
6) \( 7x^{3} - 14x^{5} = 7x^{3}(1 - 2x^{2}) \)
7) \( 3a - 3b + ax - bx = (a - b)(3 + x) \)
8) \( 17x^{2} - 17 = 17(x^{2} - 1) = 17(x - 1)(x + 1) \)
9) \( 8a^{2}b^{2} - 72a^{2}c^{2} = 8a^{2}(b^{2} - 9c^{2}) = 8a^{2}(b - 3c)(b + 3c) \)
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Bonus Knowledge
Let's dive into the wondrous world of algebraic expressions! To start, the expression \( a^{2}-9 \) is a classic example of a difference of squares. It factors neatly into \( (a-3)(a+3) \), which is not just a clever trick in algebra, but a useful tool for solving equations and simplifying calculations! Now, take a look at \( 9x^2 - 12xy + 4y^2 \). This expression can be rewritten as \( (3x - 2y)^2 \). Recognizing patterns like these helps us streamline potentially complex problems and is a key skill in both algebra and higher mathematics. So, keep your factoring skills sharp!
