Question

if a and b are the roots of x2+x+1=0 then a3+b3 is equal to​

Answer 1

The value of α³ + β³ = 2.

Step-by-step explanation:

★ Given that :

• α and β are the zeroes of the Quadratic Polynomial p(x) = x² + x + 1 = 0

★ To find :

• Value of α³ + β³ .

★ Let :

◼ The general form of the Quadratic Polynomial is : ax² + bx + c = 0.

◼ Consider the given Quadratic Polynomial.

• a = 1
• b = 1
• c = 1

➡ Sum of the zeroes : α + β = -b/a = -1/1 = 1

➡ Product of the zeroes : αβ = c/a = 1/1 = 1

Now, we know the identity :

☯ (a + b)³ = a³ + b³ + 3ab(a + b)

Consider it as

☯ (α + β)³ = α³ + β³ + 3αβ(α + β)

• Substitute the zeroes.

➠ (- 1)³ = α³ + β³ + 3(1)(-1)

➠ - 1 = α³ + β³ - 3

➠ - 1 + 3 = α³ + β³

➠ α³ + β³ = 2.

∴ The value of α³ + β³ = 2

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