Q: Prove that x^{2} – x cos(A + B) + 1 is a factor of 2x^{4} + 4x^{3} sinA sinB – x^{2}(cos2A + cos2B) + 4 x cosA cosB – 2 and also find the other factor.

Solution: Let 2x^{4} + 4x^{3} sinA sinB – x^{2} (cos2A + cos2B) + 4xcosA cosB – 2

= [x^{2} – xcos (A+B) + 1] [ax2 + bx + c]

= ax^{4} + (b – acos (A+B)) x^{3} + (a – bcos (A+B) + c) x^{2} + (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 ax^{2} + bx + c = 2x^{2} + 2cos (A – B) x – 2 = 2(x^{2} + cos(A- B)x – 1).

#### Also Read :

- If in a ΔABC , ∠B = π/3, then the maximum value of sinA sinC is
- If 4 sinA + secA = 0 then tanA equals to
- If |x| < 1 and |y| < 1 then prove that ...
- Prove that : (a+b)^p ≤ a^p + b^p ; a , b > 0 , 0 < p < 1
- If P(1) = 0 and dP(x)/dx > P(x) for all x ≥ 1 then prove that P(x) > 0 for all x > 1.
- If x = 1 + a + a^2 + a^3 + ..... to ∞ (|a| < 1), y = 1 + b + b^2 + b^3 + ..... to ∞ (|b| < 1)…
- If α + β + γ = π and tan(β+γ-α)/4 . tan(γ+α -β)/4 .tan(α+β-γ)/4 = 1. Prove that 1 + cos α + cos β +…
- Suppose that a1 , a2, ... an, an+1, ... are in A.P. and sk = a(k-1)n+1 + a(k-1)n+2 + ... akn. Prove…
- If a1, a2, a3 ........an are positive and (n – 1)s = a1 + a2 +…… + an then prove that .......
- Prove that : Sin^2 θ < sin(sinθ) for 0 < θ < π/2