use synthetic division to find the result when 2x^3+3x^2-21x+20 is divided by x-2 if there is a remainder express the result is the form of q(x)+r(x)/b(x)

Question

show your work but explain simply without using all the brackets

Answer 1

To perform synthetic division of the polynomial \( 2x^3 + 3x^2 - 21x + 20 \) by \( x - 2 \), we will first use 2 (the root of \( x - 2 = 0 \)) in the synthetic division setup.

Here are the steps:

1. Set Up the Synthetic Division:

   • Write down the coefficients of the polynomial: \( 2 \) (for \( 2x^3 \)), \( 3 \) (for \( 3x^2 \)), \( -21 \) (for \( -21x \)), and \( 20 \) (the constant term).

2. Synthetic Division Process:

   • Write the number 2 to the left and the coefficients to the right:

   2 |  2   3   -21   20
      |___________________
   

3. Bring Down the First Coefficient:

   • Bring down the 2 directly below the line:

   2 |  2   3   -21   20
      |___________________
         2
   

4. Multiply and Add:

   • Multiply 2 (the number we brought down) by 2 (the value on the left) and write the result (4) under the next coefficient (3):

   2 |  2   3   -21   20
      |      4
      |___________________
         2   7
   

   • Now, add 3 and 4 which gives 7. Write that below.

   • Next, take the result (7) and multiply it by 2, which gives 14. Write that under -21:

   2 |  2   3   -21   20
      |      4   14
      |___________________
         2   7   -7
   

   • Add -21 and 14 which gives -7.

   • Finally, multiply -7 by 2, resulting in -14, and write that under 20:

   2 |  2   3   -21   20
      |      4   14  -14
      |___________________
         2   7   -7   6
   

   • Add 20 and -14, which gives 6.

5. Interpreting the Result:

   • The numbers below the line represent the coefficients of the resulting polynomial. The coefficients are \( 2 \), \( 7 \), and \( -7 \). Thus, the resulting polynomial is:

   \[ 2x^2 + 7x - 7 \]

   • The last number (6) is the remainder.

6. Final Result:

   • Therefore, the result of dividing \( 2x^3 + 3x^2 - 21x + 20 \) by \( x - 2 \) can be expressed in the form \( q(x) + \frac{r}{b(x)} \), where:

   • \( q(x) = 2x^2 + 7x - 7 \)

   • \( r = 6 \)

   • \( b(x) = x - 2 \)

So, the final answer is:

\[ 2x^2 + 7x - 7 + \frac{6}{x - 2} \]