Problem

Given the polynomial P(x)=2x3−3x2+4x−5 and Q(x)=x−1, find the remainder when P(x) is divided by Q(x). Additionally, if R(x) is the remainder and θ is the root of Q(x), find the value of cos⁡(θ) if R(θ)=cos⁡(θ)

Answer

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Answer

Step 4: Since it's given that R(θ)=cos⁡(θ), we have cos⁡(θ)=−2

Steps

Step 1 ：Step 1: Divide P(x) by Q(x) using polynomial long division. This gives us P(x)=Q(x)⋅(2x2−x+5)+R(x), where R(x) is the remainder.

Step 2 ：Step 2: Set Q(x)=0 to find the root θ, we have x−1=0 which gives θ=1

Step 3 ：Step 3: Substitute θ=1 into R(x), we get R(1)=2⋅13−3⋅12+4⋅1−5=−2

Step 4 ：Step 4: Since it's given that R(θ)=cos⁡(θ), we have cos⁡(θ)=−2

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