Rozłóż wielomian w na czynniki A)w(x)=25x³+√5 B)w(x)=27x^6-1

[latex]mathrm{a) W(x)=25x^3+ sqrt{5}=5^2x^3+5^{ frac{1}{2} } =5^{ frac{1}{2}}(5^{ frac{3}{2} }x^3+1)= } \ mathrm{ =5^{ frac{1}{2}}( (sqrt{5} x)^3+1)=5^{ frac{1}{2}}( sqrt{5} x+1)[(sqrt{5}x)^2-sqrt{5}x+1]=} \ mathrm{ = sqrt{5}(sqrt{5}x+1)(5x^2-sqrt{5}x+1)}[/latex] [latex]mathrm{b) W(x)=27x^6-1=(3x^2)^3-1^3=} \ mathrm{ =(3x^2-1)((3x^2)^2+3x^2 +1^2)=} \ mathrm{ = ( (sqrt{3} x)^2-1^2)(9x^4+3x^2+1)=} \ mathrm{ =(sqrt{3}x-1)(sqrt{3}x+1)(9x^4+3x^2+1)}[/latex]

Korzystamy z wzorów skróconego mnożenia: a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a - b)(a² + ab + b²) a² - b² = (a + b)(a - b) [latex]A.\W(x) = 25x^{2}+sqrt{5} = 5^{2}x^{3}+5^{frac{1}{2}}=5^frac{1}{2}cdot5^frac{3}2}}x^{3}+5^{frac{1}{2}}=5^{frac{1}{2}}(5^{frac{3}{2}}x^{3}+1)=\\=5^{frac{1}{2}}((5^{frac{1}{2}}x^{3})+1^{3}))=5^{frac{1}{2}}((5^{frac{1}{2}}+1)(5^{frac{1}{2}}x)^{2}-5^{frac{1}{2}}xcdot1+1^{2}))=\\=sqrt{5}(sqrt{5}x+1)(5x^{2}-sqrt{5}x + 1)[/latex] [latex]B.\W(x) = 27x^{6}-1 = (3x^{2})^{3}-1^{3}=(3x^{2}-1)((3x^{2})^{2}+3x^{2}+1^{2}))=\\=((sqrt{3}x)^{2}-1^{2}))(9x^{4}+3x^{2}+1)=(sqrt{3}+1)(sqrt{3}x-1)(9x^{4}+3x^{2}+1)[/latex]