Question

if y^2-1 is a factor of y^4+my^3+2y^2-3y+n find value of m and n

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Answer 1

Question :–

▪︎ if y²-1 is a factor of y⁴ + my³ + 2y² - 3y + n = 0 , then find value of m and n.

ANSWER :–

$\\ \longrightarrow \: \: { \bold{ m = 3 \: \: , \: \: n = - 3 }}$

GIVEN :–

▪︎ A polynomial y⁴ + my³ + 2y² - 3y + n = 0 have a factor (y² - 1) .

TO FIND :–

▪︎ Value of m and n .

SOLUTION :–

• Given polynomial have a factor (y² - 1) , We should write this as –

$\\ \implies{ \bold{ {y}^{2} - 1 = (y + 1)(y - 1) }}$

• So that , y = -1 and y = 1 will satisfy the given polynomial.

$\\ \longrightarrow \: \: { \red{ \bold{ \underline{when \: \: y \: = \: - 1} } : - }}$

$\\ \implies \: \: { \bold{ {( - 1)}^{4} + m( - 1) ^{3} + 2 {( - 1)}^{2} - 3( - 1) + n = 0 }}$

$\\ \implies \: \: { \bold{ 1 - m + 2 + 3 + n = 0 }}$

$\\ \: \implies \: \: { \green { \bold{ n - m = - 6 \: \: \: \: - - - - eq.(1)}}}$

$\\ \longrightarrow \: \: { \red{ \bold{ \underline{when \: \: y \: = \: 1} } : - }}$

$\\ \implies \: \: { \bold{ {( 1)}^{4} + m( 1) ^{3} + 2 {( 1)}^{2} - 3( 1) + n = 0 }}$

$\\ \implies \: \: { \bold{ 1 + m + 2 - 3 + n = 0 }}$

$\\ \implies \: \: { \green { \bold{ m + n = 0 \: \: \: \: \: - - - - eq.(2) }}}$

• Now Add both equations –

$\\ \implies \: \: { \bold{ 2n = - 6 }}$

$\\ \implies \: \: { \pink{ \boxed{ \bold{ n = - 3 }}}}$

• Now put the value of n in eq.(2) –

$\\ \implies \: \: { \pink{ \boxed{ \bold{ m = 3 }}}}$