Question

Find the quadratic polynomial whose zeroes are -2 and 5. Verify the relationship
between zeroes and coefficients of the polynomial​

Answer 1

$\huge\underline{\bf{Given}}$

• Zeroes of a quadratic polynomial is -2 and 5.

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$\huge\underline{\bf{To\: Find}}$

• Quadratic polynomial.

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$\huge\underline{\bf{To\: Verify}}$

• Relationship between zeroes and coefficient of the polynomial.

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$\huge\underline{\bf{Solution}}$

• Let the zeroes of the polynomial be α and β.

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$\tt\longrightarrow{\pink{Sum\: of\: zeroes }}$

$\tt\longmapsto{α + β = (-2) + 5}$

$\tt\longmapsto{α + β = 3}$

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$\tt\longrightarrow{\pink{Product\: of\: zeroes }}$

$\tt\longmapsto{αβ = (-2) \times 5}$

$\tt\longmapsto{αβ = -10}$

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★ We know that

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$\boxed{\bf{\bigstar{Polynomial = x^2 - (α + β)x + (αβ) {\bigstar}}}}$

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★ Required Quadratic polynomial

$\tt:\implies{p(x) = x^2 - 3x + (-10)}$

$\tt:\implies{p(x) = x^2 - 3x - 10}$

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$\huge\underline{\bf{Verification}}$

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$\boxed{\bf{\bigstar{Relationship\: between\: zeroes\: and\: coefficients {\bigstar}}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\purple{Sum\: of\: zeroes = \dfrac{- coefficient\: of\: x}{coefficient\: of\: x^2}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {3 = \dfrac{-(-3)}{1}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {3 = 3}$

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$\tt:\implies\: \: \: \: \: \: \: \: {\purple{Product\: of\: zeroes = \dfrac{constant}{coefficient\: of\: x^2}}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {-10 = \dfrac{-10}{1}}$

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$\tt:\implies\: \: \: \: \: \: \: \: {10 = 10}$

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$\underline{\underline{\mathcal{\green{Hence\: verified}}}}$

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