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

1.	How to find cubic polynomial from sum of zeros ,sum of product of zeros,product of zeros
Answer» $\rule{400}4$ ☆ 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. $\rule{400}2$ ▪︎ For example:- $\sf \mapsto \: if \alpha,\: \beta \: and \: \gamma \\ \tt are \: zeroes \: of \:cubic \: polynomial.$ $\rule{400}2$ ☆ 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. $\rule{400}2$ ▪︎ 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. $\rule{400}4$ ▪︎ 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. $\rule{400}2$ ANSWER:- Here, Sum of zeroes, $\tt= 4+5+6 \\ \\ \tt = <klux>15</klux> = \frac{-b}{a}.$ 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, $\tt = (4\times5)+(5\times 6)+(6\times 4)\\ \\ = \tt 20+30+24 \\ \\ \tt = 74 = \frac{c}{a}$ $\rule{400}1$ ▪︎ So by Comp9we get, a = 1. -b = 15 =》 b = -15. c = 74. -d = 120 =》 d = -120. $\rule{400}1$ Therefore the polynomial is:- $\tt\implies ax^3+bx^2+cx+d. \\ \\ = 1(x)^3+(-15)x^2+(74)x+(-120). \\ \\ = x^3-15x^2+74x-120.$ $\rule{400}4$

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

Answer»

$\rule{400}4$

☆ 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.

$\rule{400}2$

▪︎ For example:-

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

$\rule{400}2$

☆ 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.

$\rule{400}2$

▪︎ 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.

$\rule{400}4$

▪︎ 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.

$\rule{400}2$

ANSWER:-

Here,

Sum of zeroes,

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

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,

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

$\rule{400}1$

▪︎ So by Comp9we get,

a = 1.

-b = 15 =》 b = -15.

c = 74.

-d = 120 =》 d = -120.

$\rule{400}1$

Therefore the polynomial is:-

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

$\rule{400}4$