多項式，x 位移

 

Yes! Let's perform continuous synthetic division step by step until we get the Nested Division Form of

$$3x^3 - 5x^2 - 7x - 4$$

divided by $x-3.$

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Step 1: First Synthetic Division

We divide $3x^3 - 5x^2 - 7x - 4$ using synthetic division.

Set Up the Synthetic Division Table

1. Write the coefficients of the polynomial: $$3, -5, -7, -4$$
   
2. The divisor is $x - 3$, so we use $\mathrm{3\; as\; the\; synthetic\; division\; factor.}$

$$\begin{array}{r|rrrr} 3 & 3 & -5 & -7 & -4 \\ & & 9 & 12 & 15 \\ \hline & 3 & 4 & 5 & 11 \\ \end{array}$$

• Quotient: $3x^2+4x+5$
• Remainder: $11$
  

Thus, we now divide $3x^2+4x+5.$

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Step 2: Second Synthetic Division

Now, divide $3x^2+4x+5.$

$$\begin{array}{r|rrr} 3 & 3 & 4 & 5 \\ & & 9 & 39 \\ \hline & 3 & 13 & 44 \\ \end{array}$$

• Quotient: $3x+13$
• Remainder: $44$

Now, divide $3x + 13$ by $x - 3$.

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Step 3: Third Synthetic Division

Now, divide $3x + 13$ by $x-3.$

$$\begin{array}{r|rr} 3 & 3 & 13 \\ & & 9 \\ \hline & 3 & 22 \\ \end{array}$$

• Quotient: $3$
  
• Remainder: $22$
  

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Step 4: Express in Nested Division Form

Now, we reconstruct the polynomial using the results of the synthetic divisions.

1. The final quotient is 3.

2. The previous quotient was $3x+\mathrm{13,\; which\; we\; write\; as:}$

   $$3(x - 3) + 22$$

3. The next quotient was $3x^2 + 4x + 5$, which we write using the previous step:

   $$3(x - 3)^2 + 22(x - 3) + 44$$

4. Finally, including the remainder:

   $$3(x - 3)^3 + 22(x - 3)^2 + 44(x - 3) + 11$$
   

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Final Answer:

$$\boxed{3(x - 3)^3 + 22(x - 3)^2 + 44(x - 3) + 11}$$

This is the Nested Division Form of $3x^3 - 5x^2 - 7x - 4$ when divided by $x - 3$.