# Factor Theorem-10

### From Homeworkwiki

**If (x + a) be the HCF of x ^{2} + mx + n and x^{2} + rx + s, then show that a = (n - s) / (m - r)**

**Solution:** Let P(x) = x^{2} + mx + n and Q(x) = x^{2} + rx + s,

(x + a) is the HCF of both P(x) and Q(x) means, (x + a) is a factor of both P(x) and Q(x). Therefore,

P( -a) = 0; => ( -a)^{2} + m( -a) + n = 0; => a^{2} -am + n = 0 -------------------------------(i)

Q( -a) = 0; => ( -a)^{2} + r( -a) + s = 0; => a^{2} -ar + s = 0 -------------------------------(ii)

By (i) – (ii), we get

ar – am + n – s = 0

=> am – ar = n – s

=> a(m - r) = n – s

=> a = (n - s) / (m - r)