If a and are the zeroes of the quadratic polynomial f(x) = 3x² - 5x – 2, then a³ + B³ is
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EXPLANATION.
a and b are the zeroes of the quadratic polynomial.
⇒ f(x) = 3x² - 5x - 2.
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
⇒ a + b = -(-5/3) = 5/3.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
⇒ ab = (-2/3).
To find : a³ + b³.
As we know that,
Formula of :
⇒ (x³ + y³) = (x + y)(x² - xy + y²).
⇒ (x² + y²) = (x + y)² - 2xy.
Using this formula in the equation, we get.
⇒ (a³ + b³) = (a + b)(a² - ab + b²).
⇒ (a³ + b³) = (a + b)[(a + b)² - 2ab - ab].
⇒ (a³ + b³) = (a + b)[(a + b)² - 3ab].
Put the values in the equation, we get.
⇒ (a³ + b³) = (5/3)[(5/3)² - 3(-2/3)].
⇒ (a³ + b³) = (5/3)[25/9 + 2].
⇒ (a³ + b³) = (5/3)[43/9].
⇒ (a³ + b³) = 215/27.
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