Question

if alfha , beta are the zeroes of the polynomial p(x)=3x^2-4x-4,then find the value of alfha^3+beta^3​

Answer 1

Step-by-step explanation:

Given -

• α and β are zeroes of polynomial p(x) = 3x² - 4x - 4

To Find -

• Value of α³ + β³

Now,

→ 3x² - 4x - 4

By using quadratic formula :-

• x = -b ± √b² - 4ac/2a

here,

a = 3

b = -4

c = -4

→ -(-4) ± √(-4)² - 4×3×-4/2(3)

→ 4 ± √16 + 48/6

→ 4 ± √64/6

→ 4 ± 8/6

Zeroes are -

→ x = 4 + 8/6

→ 12/6

→ 2

And

→ x = 4 - 8/6

→ -4/6

→ -2/3

Then,

The value of α³ + β³ is

→ (-2/3)³ + (2)³

→ -8/27 + 8

→ -8 + 216/27

→ 208/27

Hence,

The value of α³ + β³ is 208/27

Answer 2

Answer:

The value of ${ \alpha }^{3} \: + \: { \beta }^{3}$ is $\frac{208}{27}$

Step-by-step-explanation:

The given quadratic equation is $\sf\:3x^{2}-4x-4=0$.

Comparing with $\sf\:ax^{2}+bx+c=0$, we get,

$\sf\:a = 3,\: b = - 4,\: c = - 4$.

$\alpha\:+\:\beta\:=\:-\:\frac{\sf\:b\:}{\sf\:a}\: =\:-\:\frac{\sf\:-4}{\sf\:3}\:=\:\frac{\sf\:4}{\sf\:3}\:\:\:...(1)\\\\\alpha.\beta\:=\:\frac{\sf\:c}{\sf\:a}\:=\:\frac{\sf\:-4}{\sf\:3}\:\:\:\:...(2)$

$\sf \:Now,\:\\\\\:\red{\boxed{{\alpha}^{3}\:+\:{\beta}^{3}\:=\:{(\:\alpha\:+\:\beta\:)}^{3}\:-\:3\:\alpha.\beta\:(\:\alpha\:+\:\beta\:)}}$

$\sf\:Now, \:by\:substituting\:the\:values,\:we\:get,$

${\alpha}^{3}\:+\:{\beta}^{3}\:=\:{{(\frac{\:4}{\: 3}}\:)}^{3}\:-\:3\:\times\:(\:\frac{-4}{3}\:)\:\times\:(\frac{\:4}{\:3}\:)\:\:\:....[\:\sf\:From\:(1)\:and\:(2)\:]$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64}{27}\:-\cancel{3}\:\times\:\frac{-4}{\cancel{3}}\:\times\:\frac{4}{3}$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64}{27}\:+\:4\:\times\:\frac{4}{3}$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64}{27}\:+\:\frac{16}{3}\:\:\:....(\sf\:by\:BODMAS\:)$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64}{27}\:+\:\frac{16\times9}{3\times9}\:\:\:...(\sf\:by\:taking\:LCM\:)$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64}{27}\:+\:\frac{144}{27}$

$\therefore\:{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{64+144}{27}$

$\therefore\boxed{{\alpha}^{3}\:+\:{\beta}^{3}\:=\:\frac{208}{27}}$

Ans. : The value of ${ \alpha }^{3} \: + \: { \beta }^{3}$ is $\frac{208}{27}$

Additional Information:

1. Quadratic Equation :

An equation having a degree ‘2’ is called quadratic equation.

The general form of quadratic equation is

ax² + bx + c = 0

Where, a, b, c are real numbers and a ≠ 0.

2. Roots of Quadratic Equation:

The roots means nothing but the value of the variable given in the equation.

3. Methods of solving quadratic equation:

There are mainly three methods to solve or find the roots of the quadratic equation.

A) Factorization method

B) Completing square method

C) Formula method

4. Formula for solving quadratic equation:

$\boxed{\sf \: x \: = \: \frac{ - b \: \pm \: \sqrt{ {b}^{2} \: - \: 4ac} }{ 2a} }$