1)
36t² - 24t + 4 = (6t - 2)²
2)
(x+1)³ - (x - 2)³ = x³ + 3x² + 3x + 1 - (x³ - 6x² + 12x - 8) =
= x³ + 3x² + 3x + 1 - x³ + 6x² - 12x + 8 = 9x² - 9x + 9 =
= 9(x² - x + 1)
(x² - x + 1) nu se descompune deoarece ecuatia atasata nu are radacini reale.
3)
(x - 3)² + 5² = (x + 1)²
x² - 6x + 9 + 25 = x² + 2x + 1
x² - 6x - x² - 2x = 1 - 9 - 25
-8x = - 33 | ×(-1)
8x = 33
x = 33 / 8
x = 4,125
4)
a) DVA = R - {1; 2}
[tex]b)\,\,\, E(x)=\frac{8x-16}{ x^{2}+x+1}: \frac{ x^{2}-4x+4}{ x^{3}-1}= \\ \\ =\frac{8(x-2)}{ x^{2}+x+1}: \frac{ (x-2)^{2}}{ (x-1)( x^{2} +x+1)}= \\ \\ \frac{8(x-2)}{ x^{2}+x+1}* \frac{ (x-1)( x^{2} +x+1)}{ (x-2)^{2}} = \frac{8(x-1)}{x-2} [/tex]
[tex]c)\,\,\, E( \frac{1}{2})= \frac{8(x-1)}{x-2}=\frac{8( \frac{1}{2}-1)}{\frac{1}{2}-2}= \frac{8(-\frac{1}{2})}{-\frac{3}{2}} =(- \frac{8}{2} )*(- \frac{2}{3} )= \frac{8}{3} [/tex]
d) E(x) ∈ N daca (x - 2) este divizor al lui 8
x - 2 = 1; => x = 3
x - 2 = 2; => x = 4
x - 2 = 4; => x = 6
x - 2 = 8; => x = 10
[tex]5)\,\,\, 4^{2n} +2^{2n+1}+1 = \\ \\ = (2^{2})^ {2n}+2*2^{2n} + 1= \\ \\ =(2^{2n})^ {2}+2*2^{2n} + 1= \\ \\ (2^{2n}+1)^{2} \\ \\ cctd[/tex]
