Question

To help a poor student, ariv donated α pens and β books to him, where α and β represent the zeroes of polynomial x^2-5x + 4. inspired by the deed of ariv,arya also donated 2α+ 3β pens and 3α + 2β books to that student. find a polynomial whose zeroes are the number of pens and books donated by arya. explain why arya got inspired by the deed of ariv.​

Answer 1

Step-by-step explanation:

Given:To help a poor student, ariv donated α pens and β books to him, where α and β represent the zeroes of polynomial x²-5x + 4. inspired by the deed of ariv,arya also donated 2α+ 3β pens and 3α + 2β books to that student.

To find: find a polynomial whose zeroes are the number of pens and books donated by arya. explain why arya got inspired by the deed of ariv.

Solution:

As $\alpha$ and $\beta$ are zeros of $x^2-5x+4$

So,According to relation of zeros and coefficient of quadratic polynomial

$\alpha + \beta = 5...eq1 \\ \\ \alpha \beta = 4...eq2 \quad$

ATQ,

Arya donated $2\alpha+3\beta$ pens

and $3\alpha+2\beta$ books.

let

$\bold{m=2\alpha+3\beta}$

$\bold{n=3\alpha+2\beta}$

Quadratic polynomial with zeros m and n is given by

${x}^{2} - (m + n)x + mn$

Find Value of m+n:

$m + n = 2 \alpha + 3 \beta + 3 \alpha + 2 \beta \\ \\ m + n = 5 \alpha + 5 \beta \\ \\ m + n = 5 (\alpha + \beta)$

put value from eq1

$m + n = 5 \times 5 \\ \\ m + n = 25...eq3$

Find mn:

$mn = (2 \alpha + 3 \beta )(3 \alpha + 2 \beta ) \\ \\ mn = 6 { \alpha }^{2} + 4 \alpha \beta + 9 \alpha \beta + 6 { \beta }^{2} \\ \\ mn = 6( { \alpha }^{2} + { \beta }^{2} ) + 13 \alpha \beta...eq4$

find value of ${ \alpha }^{2} + { \beta }^{2}$

Square eq1 both sides

$( { \alpha + \beta })^{2} = 25 \\ \\ open \: identity \\ \\ { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta = 25 \\ \\ { \alpha }^{2} + { \beta }^{2} = 25 - 2 \alpha \beta \\ \\ { \alpha }^{2} + { \beta }^{2} = 25 - 2 \times 4 \\ \\ { \alpha }^{2} + { \beta }^{2} = 25 - 8 \\ \\ { \alpha }^{2} + { \beta }^{2} = 17$

put value of ${ \alpha }^{2} + { \beta }^{2}$ in eq4

$mn = 6 \times 17+13(4) \\ \\ mn = 102+52 \\ \\ mn = 154$

Therefore, quadratic polynomial is

${x}^{2} - 25x + 154$

Final answer:

The quadratic polynomial is

$\bold{\red{{x}^{2} - 25x + 154}}$

Arya got inspired by the deed of ariv because helping poor students is a good behavior.

Hope it helps you.

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