Question

Теорема Виета,помогите

Answer 1

Сумма корней равна $\frac{7}{3}$.
Произведение корней равно $\frac{1}{3}$
(x₁ + x₂)² = x₁² + 2 x₁x₂ + x₂²
x₁² + 2x₁x₂ + x₂² = (7/3)²
x₁² + x₂² = (7/3)² - 2 * 1/3 = 49/9 - 2/3 = 43/9 = 4 7/9

x₁³ + x₂³ = (x₁ + x₂)(x₁² -x₁x₂ + x₂²) = 7/3 * (4 7/9 - 1/3) = 7/3 * 4 4/9 =
= 7/3 * 40/9 = 280/27 = 10 10/27
(x₁ - x₂)² = x₁² - 2x₁x₂ + x₂² = 4 7/9 - 2 * 1/3 = 4 7/9 - 2/3 = 4 1/9
$|x _{1} -x _{2} |=| \sqrt{4 \frac{1}{9} } |=| \sqrt{ \frac{37}{9} } |=| \frac{ \sqrt{37} }{3}|$

Answer 2

$3x^2-7x+1=0$
$1) \\ x_1+x_2= \frac{7}{3} \\ \\ 2) \\ x_1x_2= \frac{1}{3}$

3)
$\left \{ {{x_1+x_2= \frac{7}{3} } \atop {x_1x_2= \frac{1}{3} }} \right.$делим первое ур-е на второе и получаем:
$\frac{1}{x_1}+ \frac{1}{x_2}=7$

$4) \\ x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2 \\ \\ x_1^2+x_2^2= \frac{49}{6}- \frac{2}{3}= \frac{49-4}{6}= \frac{15}{2} \\ \\ 5) \\ x_1^3+x_2^3=(x_1+x_2)(x_1^2-x_1x_2+x_2^2) \\ \\ x_1^3+x_2^3= \frac{7}{3}( \frac{15}{2}- \frac{1}{3})= \frac{7}{3}* \frac{43}{6} = \frac{301}{18}$