Question

find the values of p and q if x^2 - 1 is a factor of x^4 - 7x^3 + px^2 + qx - 10

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: {x}^{2} - 1 \: is \: factor \: of \: {x}^{4} - {7x}^{3} + p {x}^{2} + qx - 10$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:p \: and \: q$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: {x}^{2} - 1 \: is \: factor \: of \: {x}^{4} - {7x}^{3} + p {x}^{2} + qx - 10$

Let assume that

$\rm :\longmapsto\: f(x)\: = \: {x}^{4} - {7x}^{3} + p {x}^{2} + qx - 10$

Now, further it is given that

$\rm :\longmapsto\: {x}^{2} - 1 \: is \: factor \:of \: f(x)$

$\rm :\longmapsto\: (x - 1)(x + 1) \: is \: factor \:of \: f(x)$

Two cases arises :-

Case 1

$\rm :\longmapsto\: x - 1 \: is \: factor \:of \: f(x)$

We know,

By Factor Theorem, it state that if a polynomial p(x) is divisible by linear polynomial (x - a), then p(a) = 0 or if linear polynomial (x - a) is a factor of polynomial p(x), then p(a) = 0.

Thus,

$\rm :\longmapsto\:f(1) = 0$

$\rm :\longmapsto\: {(1)}^{4} - {7(1)}^{3} + p {(1)}^{2} + q(1) - 10 = 0$

$\rm :\longmapsto\:1 - 7 + p + q - 10 = 0$

$\rm :\longmapsto\:+ p + q - 16 = 0$

$\bf :\longmapsto\:+ p + q = 16 - - - - (1)$

Case - 2

$\rm :\longmapsto\: x + 1 \: is \: factor \:of \: f(x)$

So, by factor theorem,

$\rm :\longmapsto\:f( - 1) = 0$

$\rm :\longmapsto\: {( - 1)}^{4} - {7( - 1)}^{3} + p {( - 1)}^{2} + q( - 1) - 10 = 0$

$\rm :\longmapsto\:1 + 7 + p - q - 10 = 0$

$\rm :\longmapsto\: p - q - 2= 0$

$\bf :\longmapsto\: p - q = 2 - - - - (2)$

Now, On adding equation (1) and (2), we get

$\rm :\longmapsto\:2p = 18$

$\bf :\longmapsto\:p = 9 - - - - - (3)$

On substituting value of p in equation (1), we get

$\rm :\longmapsto\:9 + q = 16$

$\rm :\longmapsto\: q = 16 - 9$

$\bf :\longmapsto\: q = 7$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: p = 9}} \: \: \: and \: \: \: \underbrace{ \boxed{ \bf \: q = 7}}$

Additional Information :-

Remainder Theorem :-

This theorem states that if a polynomial p(x) is divisible by linear polynomial (x - a), then remainderis p(a).