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Use the remainder theorem to find P(3)P(3) for P(x)=2x3−4x2−2x+9P(x)=2x^{3}-4x^{2}-2x+9. Specifically, give the quotient and the remainder for the associated division and the value of P(3)P(3). Quotient : □\square Remainder =□□=\square \square P(3)=P(3)= □\square

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Q. Use the remainder theorem to find P(3)P(3) for P(x)=2x3−4x2−2x+9P(x)=2x^{3}-4x^{2}-2x+9. Specifically, give the quotient and the remainder for the associated division and the value of P(3)P(3). Quotient : □\square Remainder =□□=\square \square P(3)=P(3)= □\square
  1. Apply Remainder Theorem: Step 11: Apply the remainder theorem to find P(3)P(3). The remainder theorem states that the remainder of the division of a polynomial P(x)P(x) by (x−c)(x - c) is P(c)P(c). Here, c=3c = 3. Calculate P(3)=2(3)3−4(3)2−2(3)+9P(3) = 2(3)^3 - 4(3)^2 - 2(3) + 9. = 2(27)−4(9)−6+92(27) - 4(9) - 6 + 9 = 54−36−6+954 - 36 - 6 + 9 = 2121
  2. Perform Synthetic Division: Step 22: Perform synthetic division to find the quotient and remainder.\newlineDivide P(x)P(x) by (x−3)(x - 3) using synthetic division.\newlineCoefficients of P(x)P(x): 2,−4,−2,92, -4, -2, 9\newlineUsing 33 as the synthetic divisor:\newline\begin{array}{cccc}\(\newline2 & | & 6 & | & 6 & | & 12 (\newline\)\hline\newline2 & | & 2 & | & 4 & | & 21\newline\end{array}\)\newlineQuotient: 2x2+2x+42x^2 + 2x + 4\newlineRemainder: 2121

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