Question

) If α and β are zeros of the polynomial p(x) =3x2

-10x+7 then find the value of α2+β2
​

Answer 1

Step-by-step explanation:

Given -

• α and β are zeroes of the polynomial p(x) = 3x² - 10x + 7

To Find -

• Value of α² + β²

Now,

→ 3x² - 10x + 7

→ 3x² - 3x - 7x + 7

→ 3x(x - 1) - 7(x - 1)

→ (3x - 7)(x - 1)

Zeroes are -

3x - 7 = 0 and x - 1 = 0

• x = 7/3 and x = 1

Then,

The value of α² + β² is

→(7/3)² + (1)²

→ 49/9 + 1

→ 49 + 9/9

→ 58/9

Hence,

The value of α² + β² is 58/9

Answer 2

Answer :

Given :

$\tt{p(x) = 3 {x}^{2} - 10x + 7 }$

$\rule{400}{4}$

Required to Find :

$\tt{1. \: \: \: \: \: \: \: \: \: \: \: \: { \alpha }^{2} + { \beta }^{2} }$

$\rule{400}{4}$

Mentioned condition :

If α and β are zeros of the polynomial p(x)

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Explanation :

In the question it is given that α and β are zeros of the polynomial p(x) .

He asked us to find the values of α^2+β^2 .

To find the solution for the above sum we need to learn the concept behind this question .

So, let's start .

$\rule{400}{2}$

What is zero of the polynomial ?

The word zero of the polynomial , actually states that when ever this value is substituted in the polynomial in place of x then the remainder comes to be as zero .

For example : 2 is the zero of p(x) = x^3 - 8 .

so, if we substitute this 2 value in place of x

p(2) = (2)^3 - 8

= 8 - 8

= 0

Hence, 2 is the zero of the polynomial .

Now, coming to the question it is given that α and β are zeros of the polynomial .

Here, α and β are two unknown values .

We need to find the values of α and β in order to Solve this question .

The value of α and β can be found by the factorisation of the given expression .

$\rule{400}{2}$

What is factorisation ?

Factorisation is a method of expanding or reducing the size of the expression into it's factor .

So, here we need to factorise p(x) = 3x^2 - 10 x + 7

Here, consider the last term i.e 7 which is a constant .

As 7 is a prime number .

Here, we have to use the method of splitting the middle term in order to factorise this expression .

After factorisation you will be left with the values of α and β .

Hence, you can find the value of α^2 and β^2 .

Now, let's crack the question .

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Solution :

Given Quadratic expression ;

$\tt{p(x) = 3 {x}^{2} - 10x + 7 }$

If α and β are zeros of the polynomial .

Now, let's factorise the above expression .

The method used is splitting the middle term.

So,

$\tt{3{x}^{2} - 10x + 7}$

$\tt{3{x}^{2}}{\red{-7x - 3x}}{+7}$

(here the splitting is showing in red )

$\tt{ x ( 3x -7) - 1(3x - 7)}$

$\tt{\blue{ ( x - 1) ( 3x - 7)}}$

$\tt{\orange{Value\; of \; \alpha = }}$

$\longrightarrow{\tt{x - 1 = 0}}$

$\Rightarrow{\tt{x = 1}}$

$\tt{\implies{\green{value\;of\;\alpha = 1 }}}$

Similarly,

$\tt{\orange{Value\; of \; \beta = }}$

$\longrightarrow{\tt{3x - 7 = 0}}$

$\Rightarrow{\tt{3x = 7}}$

$\Rightarrow{\tt{x = \dfrac{7}{3}}}$

$\tt{\implies{\green{value\;of\;\beta = \dfrac{7}{3}}}}$

Required to find value of α2+β2

Hence, substitute their respective values !

$\Rightarrow{\tt{ {(\alpha)}^{2} + {(\beta)}^{2}}}$

$\Rightarrow{\tt{{(1)}^{2} + {(\dfrac{7}{3})}^{2}}}$

$\Rightarrow{\tt{ 1 + \dfrac{49}{9}}}$

$\Rightarrow{\tt{ \dfrac{9 + 49}{9}}}$

$\implies{\tt{ \dfrac{58}{9}}}$

$\yellow{\boxed{\tt{\therefore{Value \;of \;{\alpha}^{2}\;\; and \;\;{\beta}^{2} = \dfrac{58}{9}}}}}$

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✅ Hence Solved .