Question

How to find cubic polynomial from sum of zeros ,sum of product of zeros,product of zeros

Answer 1

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☆ ANSWER:-

• You can find out the cubic polynomial from sum of zeros ,sum of product of zeros,product of zeros by using the relationship between the zeroes and coefficients.

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▪︎ For example:-

$\sf \mapsto \: if \alpha,\: \beta \: and \: \gamma \\ \tt are \: zeroes \: of \:cubic \: polynomial.$

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☆ Then,

• Sum Of zeroes,

$\tt\leadsto \alpha + \beta + \gamma = \frac{ - b}{a}$

• Product Of zeroes,

$\tt\leadsto \alpha \beta \gamma = \frac{ - d}{a}$

• Sum of product of zeroes taken two at a time,

$\tt\leadsto (\alpha \beta +( \beta \gamma ) + ( \gamma \alpha ) = \frac{c}{a}$

=》 Where,

• a = Coefficient of x³
• -b = -Coefficient of x².
• c = Coefficient of x.
• -d = -Constant Term.

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▪︎ So,

▪︎ The Cubic Polynomial we get after seeing the relationship between the zeroes and coefficients is:-

$\tt\mapsto a {x}^{3} + b {x}^{2} + cx + d$

▪︎ Then by obtaining the values of Sum and product of the zeroes we can find the required cubic polynomial as the zeroes were already given.

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▪︎ Lets Solve an question so that you can understand betterly.

Q:- Given that 4,5 and 6 are the zeroes of a cubic polynomial find out the polynomial.

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ANSWER:-

Here,

Sum of zeroes,

$\tt= 4+5+6 \\ \\ \tt = 15 = \frac{-b}{a}.$

Product Of zeroes,

$\tt 4\times 5\times 6 \\ \\ = \tt20\times 6 \\ \\ \tt = 120 = \frac{-d}{a}$

Product of zeroes taken two at a time,

$\tt = (4\times5)+(5\times 6)+(6\times 4)\\ \\ = \tt 20+30+24 \\ \\ \tt = 74 = \frac{c}{a}$

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▪︎ So by Comp9we get,

• a = 1.

• -b = 15 =》 b = -15.

• c = 74.

• -d = 120 =》 d = -120.

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Therefore the polynomial is:-

$\tt\implies ax^3+bx^2+cx+d. \\ \\ = 1(x)^3+(-15)x^2+(74)x+(-120). \\ \\ = x^3-15x^2+74x-120.$

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