Q: Prove that x2 – x cos(A + B) + 1 is a factor of 2x4 + 4x3 sinA sinB – x2(cos2A + cos2B) + 4 x cosA cosB – 2 and also find the other factor.
Solution: Let 2x4 + 4x3 sinA sinB – x2 (cos2A + cos2B) + 4xcosA cosB – 2
= [x2 – xcos (A+B) + 1] [ax2 + bx + c]
= ax4 + (b – acos (A+B)) x3 + (a – bcos (A+B) + c) x2 + (b – c cos (A+B)) x + c
By comparing co-efficients,
a = 2, c = -2 and
b – a cos (A+B) = 4 sinA sinB …(1)
a – b cos (A+B) + c = -cos 2A – cos2B …(2)
b – c cos (A+B) = 4cosA cosB …(3)
solving, we get same value of b
b = 2cos (A – B),
hence the assumption is correct and by solving for the other factor ax2 + bx + c = 2x2 + 2cos (A – B) x – 2 = 2(x2 + cos(A- B)x – 1).
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