Math question 10 points?

Learn : If α , β , γ are the roots of cubic equation ax³ + bx² + cx + d = 0

then,

....α + β + γ = -(b/a)

....αβ + βγ + γα = (c/a)

Now consider ,

x³ - 7x² + 2x - 3 = 0 , whose roots are α , β , γ .

so ,

....α + β + γ = -(-7/1) = 7 _______

....αβ + βγ + γα = 2/1 = 2 _______

• We know that ,

(α + β + γ)² = (α² + β² + γ²) + 2(αβ + βγ + γα)

so ,

....(α² + β² + γ²) = (α + β + γ)² - 2(αβ + βγ + γα)

....(α² + β² + γ²) = 7² - 2 × 2 = 49 - 4 = 45

....(α² + β² + γ²) = 45_________Ans. (i)

• Since α , β , γ are roots of the given cubic equation so they must satisfy the given cubic equation .

....α ³ - 7α² + 2α - 3 = 0 ______(1)

....β³ - 7β² + 2β - 3 = 0 ______(2)

....γ³ - 7γ² + 2γ - 3 = 0 ______(3)

Apply (1) + (2) + (3)

....(α ³ - 7α² + 2α - 3) + (β³ - 7β² + 2β - 3) +

(γ³ - 7γ² + 2γ - 3) = 0 + 0 + 0

....(α ³ + β³ + γ³) - 7 ( α² + β² + γ² ) +

2 (α + β + γ) - 9 = 0

....(α ³ + β³ + γ³) - 7 × 45 + 2 × 7 - 9 = 0

....(α ³ + β³ + γ³) = 315 - 14 + 9 = 310

....(α ³ + β³ + γ³) = 310________Ans. (ii)