Задание 64. Разложите двучлен на множители:

Задание 64. Разложение разности квадратов на множители.

Для решения этих примеров будем использовать формулу разности квадратов: \( a^2 - b^2 = (a-b)(a+b) \). Применим её к каждому выражению.

Решение:

• 1) \( x^2 - y^2 = (x-y)(x+y) \)
• 2) \( a^2 - h^2 = (a-h)(a+h) \)
• 3) \( c^2 - 3^2 = (c-3)(c+3) \)
• 4) \( 4^2 - p^2 = (4-p)(4+p) \)
• 5) \( 25 - b^2 = 5^2 - b^2 = (5-b)(5+b) \)
• 6) \( n^2 - 1 = n^2 - 1^2 = (n-1)(n+1) \)
• 7) \( m^2 - (3x)^2 = (m-3x)(m+3x) \)
• 8) \( (2a)^2 - y^2 = (2a-y)(2a+y) \)
• 9) \( 16k^2 - 25 = (4k)^2 - 5^2 = (4k-5)(4k+5) \)
• 10) \( 100 - 49p^2 = 10^2 - (7p)^2 = (10-7p)(10+7p) \)
• 11) \( 36h^2 - 1 = (6h)^2 - 1^2 = (6h-1)(6h+1) \)
• 12) \( 0,64m^2 - 1 = (0.8m)^2 - 1^2 = (0.8m-1)(0.8m+1) \)
• 13) \( 0,09x^2 - 81y^2 = (0.3x)^2 - (9y)^2 = (0.3x-9y)(0.3x+9y) \)
• 14) \( 1 - 400y^2 = 1^2 - (20y)^2 = (1-20y)(1+20y) \)
• 15) \( 49h^2 - 0,04c^2 = (7h)^2 - (0.2c)^2 = (7h-0.2c)(7h+0.2c) \)
• 16) \( 0,01p^2 - 900t^2 = (0.1p)^2 - (30t)^2 = (0.1p-30t)(0.1p+30t) \)
• 17) \( 121s^2 - 225a^2 = (11s)^2 - (15a)^2 = (11s-15a)(11s+15a) \)
• 18) \( 1,44q^2 - 25c^2 = (1.2q)^2 - (5c)^2 = (1.2q-5c)(1.2q+5c) \)
• 19) \( 196m^2 - 0,36n^2 = (14m)^2 - (0.6n)^2 = (14m-0.6n)(14m+0.6n) \)
• 20) \( x^2y^2 - 4 = (xy)^2 - 2^2 = (xy-2)(xy+2) \)
• 21) \( 16a^4 - 9b^2 = (4a^2)^2 - (3b)^2 = (4a^2-3b)(4a^2+3b) \)
• 22) \( 36p^6 - n^2m^{10} = (6p^3)^2 - (nm^5)^2 = (6p^3-nm^5)(6p^3+nm^5) \)
• 23) \( 169k^4 - 4p^6 = (13k^2)^2 - (2p^3)^2 = (13k^2-2p^3)(13k^2+2p^3) \)
• 24) \( 0,09a^2 - 100b^4 = (0.3a)^2 - (10b^2)^2 = (0.3a-10b^2)(0.3a+10b^2) \)
• 25) \( \frac{1}{4}x^2 - \frac{4}{9}y^2 = (\frac{1}{2}x)^2 - (\frac{2}{3}y)^2 = (\frac{1}{2}x-\frac{2}{3}y)(\frac{1}{2}x+\frac{2}{3}y) \)
• 26) \( \frac{25}{64}n^2 - \frac{1}{9}m^2 = (\frac{5}{8}n)^2 - (\frac{1}{3}m)^2 = (\frac{5}{8}n-\frac{1}{3}m)(\frac{5}{8}n+\frac{1}{3}m) \)
• 27) \( \frac{4}{49}k^2 - \frac{1}{36}p^2 = (\frac{2}{7}k)^2 - (\frac{1}{6}p)^2 = (\frac{2}{7}k-\frac{1}{6}p)(\frac{2}{7}k+\frac{1}{6}p) \)
• 28) \( 0,64x^2 - \frac{1}{64}y^2 = (0.8x)^2 - (\frac{1}{8}y)^2 = (0.8x-\frac{1}{8}y)(0.8x+\frac{1}{8}y) \)
• 29) \( -1 + 4p^{100} = 4p^{100} - 1 = (2p^{50})^2 - 1^2 = (2p^{50}-1)(2p^{50}+1) \)
• 30) \( -36v^2 + \frac{1}{4}q^{10} = \frac{1}{4}q^{10} - 36v^2 = (\frac{1}{2}q^5)^2 - (6v)^2 = (\frac{1}{2}q^5-6v)(\frac{1}{2}q^5+6v) \)
• 31) \( a^{10}b^{10} - c^{10}e^{20} = (a^5b^5)^2 - (c^5e^{10})^2 = (a^5b^5 - c^5e^{10})(a^5b^5 + c^5e^{10}) \)
• 32) \( -0,64k^{10} + 25n^{14} = 25n^{14} - 0,64k^{10} = (5n^7)^2 - (0.8k^5)^2 = (5n^7 - 0.8k^5)(5n^7 + 0.8k^5) \)