Question

f(x) = x^3+x+1 , roots of g(x) are the squares of roots of f(x) , Find g(9)​

Answer 1

✪ANSWER✪

• $\red{\boxed{\green{\boxed{\blue{g(9) = 899}}}}}$

★EXPLAINATION★

☆GIVEN☆

• f(x) = x³ + x + 1

• Roots of g(x) = 0 are squares of roots of f(x) = 0

☆TO FIND☆

• Value of g(9).

☆CALCULATION☆

Let $\alpha , \beta , \gamma$ are root of f(x) = 0. Then ${\alpha}^{2} , {\beta}^{2} , {\gamma}^{2}$ are the roots of g(x) = 0.

᯽ʀᴇʟᴀᴛɪᴏɴ ʙᴇᴛᴡᴇᴇɴ ʀᴏᴏᴛs ᴀɴᴅ ᴄᴏ-ᴇғғɪᴄɪᴇɴᴛ᯽

For the equation f(x) = 0

1. sᴜᴍ ᴏғ ʀᴏᴏᴛs

$\alpha + \beta + \gamma = - \frac{coeff \: of \: {x}^{2} }{coeff \: of \: {x}^{3}}$

$⇒\alpha + \beta + \gamma = 0........(1)$

2. sᴜᴍ ᴏғ ʀᴏᴏᴛs ᴛᴀᴋᴇɴ ᴛᴡᴏ ᴀᴛ ᴀ ᴛɪᴍᴇ

$\alpha \beta + \beta \gamma + \gamma \alpha = \frac{coeff \: of \: x}{coeff \: of \: x {}^{3} }$

$⇒ \alpha \beta + \beta \gamma + \gamma \alpha = 1.......(2)$

3. ᴘʀᴏᴅᴜᴄᴛ ᴏғ ʀᴏᴏᴛs

$\alpha \beta \gamma = - \frac{constant \: term}{coeff \: of \: {x}^{3} }$

$⇒\alpha \beta \gamma = - 1......(3)$

Now, using these equations, we will calculate i) sum of roots, ii) sum of the roots taken two at a time, iii) product of roots of g(x) = 0.

✰✰✰

☞︎︎︎ (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)

☞︎︎︎ a² + b² + c² = (a+b+c)² - 2(ab + bc + ca)

For the equation g(x) = 0

i) sᴜᴍ ᴏғ ʀᴏᴏᴛs

${ \alpha }^{2} + { \beta }^{2} + { \gamma }^{2}$

$=( \alpha + \beta + \gamma ) {}^{2} -2( \alpha \beta + \beta \gamma + \gamma \alpha )$

$= {0}^{2} - 2 \times 1$

$= - 2$

ii) sᴜᴍ ᴏғ ʀᴏᴏᴛs ᴛᴀᴋᴇɴ ᴛᴡᴏ ᴀᴛ ᴀ ᴛɪᴍᴇ

${ \alpha }^{2} { \beta }^{2} + { \beta }^{2} { \gamma }^{2} + { \gamma}^{2} { \alpha }^{2}$

$= { (\alpha \beta )}^{2} + {( \beta \gamma )}^{2} + {( \gamma \alpha )}^{2}$

$= ( \alpha \beta + \beta \gamma + \gamma \alpha ) {}^{2} - 2( \alpha \beta {}^{2} \gamma + \beta \gamma {}^{2} \alpha + \gamma \alpha {}^{2} \beta )$

$= {1}^{2} - 2 \alpha \beta \gamma ( \beta + \gamma + \alpha )$

$= 1 - 2 \times ( - 1) \times 0$

$= 1$

iii) ᴘʀᴏᴅᴜᴄᴛ ᴏғ ʀᴏᴏᴛs

${ \alpha }^{2} { \beta }^{2} { \gamma }^{2}$

$= {( \alpha \beta \gamma )}^{2}$

$= {( - 1)}^{2}$

$= 1$

✫EXPRESSION OF g(x), VALUE OF g(9)✫

Now,

g(x) = x³ - (sᴜᴍ ᴏғ ʀᴏᴏᴛs)x² + (sᴜᴍ ᴏғ ʀᴏᴏᴛs ᴛᴀᴋᴇɴ ᴛᴡᴏ ᴀᴛ ᴀ ᴛɪᴍᴇ)x -(ᴘʀᴏᴅᴜᴄᴛ ᴏғ ʀᴏᴏᴛs)

$⇒g(x) = x³ - (-2)x² + (1)x - (1)$

$⇒\boxed{g(x) = x³ + 2x² + x - 1}$

$⇒g(9) = 9³ + 2.9² + 9 - 1$

$⇒\boxed{g(9) = 899}$