MHB What is the Polynomial P(x) Given a Specific Quotient and Remainder?

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To find the polynomial P(x), use the formula P = AQ + R, where A is the divisor (2x + 1), Q is the quotient (x^2 - x + 2), and R is the remainder (5). Substitute these values into the equation to get P(x) = (2x + 1)(x^2 - x + 2) + 5. Expand the product and simplify to derive the complete polynomial. The final result will provide the expression for P(x). This method effectively utilizes the relationship between polynomials, quotients, and remainders.
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I can't seem to figure this out.

When a polynomial P(x) is divided by (2x+1) the quotient is x^2-x+2 and the remainder is 5. What is P(x)?
 
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A hint: When 7 is divided by 3 the quotient is 2 and the remainder is 1.
 
If P divided by A has quotient Q with remainder R, P/A= Q+ R/A, then P= AQ+ R. You are told what A, Q, and R are. Just do the algebra to find P.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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