Question

verify that-1,-2- are the zeros of a cubic polynomial 2x3+7x2+7x+2 and then verify the relationship between the zeros and the coefficients​

Answer 1

Correct question

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Verify that -1,-2,-1/2 are the zeros of a cubic polynomial 2x³+7x²+7x+2 and then verify the relationship between the zeros and the coefficients

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A n s w e r

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G i v e n

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• Zeroes of a cubic polynomial are -1,-2,-1/2

• The cubic polynomial is 2x³+7x²+7x+2

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F i n d

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• Verify the zeroes of cubic polynomial and relation between the zeros and the coefficients

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S o l u t i o n

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$\underline{\bold{\texttt{Verification for -1 :}}}$

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If -1 is a zero of the given cubic polynomial then it will give the result 0 at the end when the variable 'x' is replaced by it

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So,

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➜ 2x³ + 7x² + 7x + 2

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➜ 2(-1)³ + 7(-1)² + 7(-1) + 2

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➜ 2(-1) + 7(1) + 7(-1) + 2

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➜ -2 + 7 - 7 + 2

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➜ 0

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• Thus -1 is a zero of 2x³ + 7x² + 7x + 2

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$\underline{\bold{\texttt{Verification for -2 :}}}$

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If -2 is a zero of the given cubic polynomial then it will give the result 0 at the end when the variable 'x' is replaced by it

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So,

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➜ 2x³ + 7x² + 7x + 2

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➜ 2(-2)³ + 7(-2)² + 7(-2) + 2

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➜ 2(-8) + 7(4) + 7(-2) + 2

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➜ -16 + 28 - 14 + 2

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➜ 30 - 30

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➜ 0

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• Thus -2 is a zero of 2x³ + 7x² + 7x + 2

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$\underline{\bold{\texttt{Verification for \dfrac { -1 } { 2 } :}}}$

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If $\sf \dfrac { -1 } { 2 }$ is a zero of the given cubic polynomial then it will give the result 0 at the end when the variable 'x' is replaced by it

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So,

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➜ 2x³ + 7x² + 7x + 2

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➜ $\sf 2\bigg( \dfrac { -1 } { 2 } \bigg) ^3 + 7\bigg( \dfrac { -1 } { 2 } \bigg) ^2 + 7 \bigg( \dfrac { -1 } { 2 } \bigg) + 2$

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➜ $\sf 2\bigg( \dfrac { -1 } { 8 } \bigg) + 7\bigg( \dfrac { 1 } { 4 } \bigg) + 7 \bigg( \dfrac { -1 } { 2 } \bigg) + 2$

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➜ $\sf \dfrac { -1 } { 4 } + \bigg( \dfrac { 7 } { 4 } - \dfrac { 7 } { 2 } \bigg) + 2$

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➜ $\sf \bigg(\dfrac{ - 1 + 7 - 14 + 8 } { 4 } \bigg)$

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➜ $\sf \bigg(\dfrac { -15 + 15} { 4 } \bigg)$

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➜ $\sf \bigg(\dfrac { 0} { 4 } \bigg)$

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➜ 0

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• Thus $\sf \dfrac { -1 } { 2 }$ is a zero of 2x³ + 7x² + 7x + 2

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Relationship between zeroes and coefficient

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$\underline{\bold{\texttt{For general cubic polynomial :}}}$

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• For a cubic polynomial ax³ + bx² + cx + d

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$\underline{\bold{\texttt{Sum of zeroes :}}}$

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➠ $\bf \alpha + \beta + \gamma = -\dfrac { b } { a }$ ⚊⚊⚊⚊ ⓵

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$\underline{\bold{\texttt{Product of zeroes :}}}$

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➠ $\bf \alpha \times \beta \times \gamma = -\dfrac { d} { a }$ ⚊⚊⚊⚊ ⓶

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$\underline{\bold{\texttt{Sum of products of zeroes :}}}$

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➠ $\bf \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac { c} { a }$ ⚊⚊⚊⚊ ⓷

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Where,

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• α = 1st zero

• β = 2nd zero

• γ = 3rd zero

• a = Coefficient of x³

• b = Coefficient of x²

• c = Coefficient of x

• d = Constant term

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$\underline{\bold{\texttt{For the given cubic polynomial :}}}$

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• For the cubic polynomial 2x³ + 7x² + 7x + 2

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$\underline{\bold{\texttt{Value Section :}}}$

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• α = -1

• β = -2

• γ = $\sf \dfrac { -1 } { 2 } = -0.5$

• a = 2

• b = 7

• c = 7

• d = 2

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$\underline{\bold{\texttt{Sum of zeroes :}}}$

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⟮ Putting values from Value Section to ⓵ ⟯

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➜ $\bf \alpha + \beta + \gamma = -\dfrac { b } { a }$

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➜ $\sf -1 + -2 - 0.5 = -\dfrac { 7} { 2}$

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➜ -3.5 = -3.5

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Verified

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$\underline{\bold{\texttt{Product of zeroes :}}}$

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⟮ Putting values from Value Section to ⓶ ⟯

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➜ $\bf \alpha \times \beta \times \gamma = -\dfrac { d} { a }$

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➜ $\sf -1 \times -2 \times -0.5 = -\dfrac { 2} { 2}$

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➜ -1 = -1

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Verified

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$\underline{\bold{\texttt{Sum of products of zeroes :}}}$

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⟮ Putting values from Value Section to ⓷ ⟯

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➜ $\bf \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac { c} { a }$

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➜ $\footnotesize{ \sf [-1(-2)] + [-2(-0.5)] + [-0.5(-1)] = \dfrac { 7} { 2}}$

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➜ 2 + 1 + 0.5 = 3.5

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➜ 3.5 = 3.5

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Verified