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Answer 1

(9)

The given polynomial is

p(x) = 4x² - 5x - 1

Since α and β are the zeroes of p(x), by the relation between zeroes and coefficients, we get

• α + β = - (- 5/4) = 5/4
• αβ = - 1/4

Then, α²β + αβ²

= αβ (α + β)

= (- 1/4) (5/4)

= - 5/16

⇒ α²β + αβ² = - 5/16

(10)

The given polynomial is

f(t) = t² - 4t + 3

Since α and β are the zeroes of f(t), by the relation between zeroes and coefficient, we get

• α + β = - (- 4/1) = 4
• αβ = 3/1 = 3

Then, α⁴β³ + α³β⁴

= α³β³ (α + β)

= (αβ)³ (α + β)

= 3³ (4)

= 27 * 4

= 108

⇒ α⁴β³ + α³β⁴ = 108

Answer 2

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