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If (x+1) is a factor of 2x^(3)+ax^(2)+2b...

If (x+1) is a factor of `2x^(3)+ax^(2)+2bx+1`, then find the value of a and b given that 2a-3b=4.

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To solve the problem, we need to find the values of \( a \) and \( b \) given that \( (x + 1) \) is a factor of the polynomial \( 2x^3 + ax^2 + 2bx + 1 \) and that \( 2a - 3b = 4 \). ### Step 1: Use the Factor Theorem Since \( (x + 1) \) is a factor of the polynomial, substituting \( x = -1 \) into the polynomial should yield zero. \[ P(-1) = 2(-1)^3 + a(-1)^2 + 2b(-1) + 1 = 0 \] ...
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