Question

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

LCM	GCD	p(x)	q(x)
a3 – 10a2 + 11a + 70	a – 7	a2 – 12a + 35

Answer 1

L.C.M. = a³ – 10a² + 11a + 70

= (a – 7) (a² – 3a – 10)

= (a – 7) (a – 5) (a + 2)

7	1 –  10 + 11 + 70 0     7  –  21  – 70
	1  –3    –10      0

G.C.D. = (a – 7)

p(x) = a² - 12a + 35

= (a – 5)(a – 7)

q(x) = `("L""C""M" xx "G""C""D")/("p"(x))`

= `((a^3-10a^2+11a+70)(a-7))/(a^2-12a+35)`

= a² - 12a + 35

= a² - 7a - 5a + 35

= a(a - 7) -5(a - 7)

= (a - 7) (a - 5)

= `((a^3 - 10a^2 + 11a + 70) (a - 7))/((a-7)(a-5))`

= `(a^3 - 10a^2 + 11a + 70) /(a-5)`

q(x) = a² - 5a - 14