Question

If the zeroes of the polynomial x cube +3xsquare+x+1 are a_b,a,a+b find a and b

Answer 1

$\large{\underline{\underline{\textsf{\maltese\: {\red{Given :-}}}}}}$

❖ Cubic Polynomial = x³ - 3x² + x + 1

❖ Zeros of the Cubic Polynomial = (a - b) , a , (a + b)

$\large{\underline{\underline{\textsf{\maltese\: {\red{To Find :-}}}}}}$

❖ Value of a and b = ?

$\large{\underline{\underline{\textsf{\maltese\: {\red{Concept Implemented :-}}}}}}$

For a cubic polynomial ,

ax³ + bx² + cx + d

Whose zeros are α , β and γ. Then ,

☯︎ Sum of zeros = α + β + γ = $\sf\dfrac{-b}{a}$

☯︎ Sum of the product of two convective zeros = αβ + βγ + γα = $\sf\dfrac{c}{a}$

☯︎ Product of zeros = αβγ = $\sf\dfrac{-d}{a}$

$\large{\underline{\underline{\textsf{\maltese\: {\red{Solution :-}}}}}}$

Comparing the Cubic Polynomial (x³ + 3x² + x + 1) with the standard form of Cubic Polynomial (ax³ + bx² + cx + d) we get,

❃ a = 1

❃ b = -3

❃ c = 1

❃ d = 1

❃ α = (a - b)

❃ β = a

❃ γ = (a + b)

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☯︎ Sum of zeros = α + β + γ = $\sf\dfrac{-b}{a}$

❂ α + β + γ = (a - b) + a + (a + b)

❂ α + β + γ = a - b + a + a + b

❂ α + β + γ = 3a⠀⠀------【1】

❂ α + β + γ = $\sf\dfrac{-b}{a}$

❂ α + β + γ = $\sf\dfrac{-(-3)}{1}$

❂ α + β + γ = 3 ⠀------【2】

From Equation 【1】and 【2】. We get,

⇒ 3a = 3

⇒ a = $\sf\dfrac{-3}{3}$

⇒ a = 1

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☯︎ Product of zeros = αβγ = $\sf\dfrac{-d}{a}$

❂ αβγ = (a - b) × a × (a + b)

❂ αβγ =⠀(a² - b²)a⠀[(a + b)(a - b) = a² - b²]

❂ αβγ = (a² - b²)a

❂ αβγ = {(1)² - b²}1⠀‎‏‏‏‏⠀‎‏‏[Putting the value of a that we've obtained]

❂ αβγ = (1 - b²)1

❂ αβγ = 1 - b²⠀⠀------‏‏【3】

❂ αβγ = $\sf\dfrac{-d}{a}$

❂ αβγ = $\sf\dfrac{-1}{1}$

❂ αβγ = -1 ⠀⠀‎‏‏⠀⠀------【4】

From Equation 【3】and 【4】. We get,

⇒ 1 - b² = -1

⇒ -b² = -1 - 1

⇒ -b² = -2

⇒ b = ±√2

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∴ a = 1

∴ b = ±√2