Use synthetic division to divide the polynomial 3x4+2x2−5x+6

Question

by x−1
.

Be sure to show all of your steps for the show work problems. You may type your answers in the answer box using or upload a

Answer 1

To use synthetic division to divide the polynomial \( 3x^4 + 0x^3 + 2x^2 - 5x + 6 \) by \( x - 1 \), we first note the root of the divisor \( x - 1 \), which is \( 1 \).

Step 1: Write down the coefficients

Write down the coefficients of the polynomial:

• Coefficient of \( x^4 \): 3
• Coefficient of \( x^3 \): 0 (since there is no \( x^3 \) term)
• Coefficient of \( x^2 \): 2
• Coefficient of \( x^1 \): -5
• Constant term: 6

Thus, the coefficients are: \[ 3, 0, 2, -5, 6 \]

Step 2: Set up Synthetic Division

We set up the synthetic division using the root \( 1 \).

1 |  3   0   2   -5   6
   |       

Step 3: Perform the synthetic division

1. Bring down the first coefficient (3):

1 |  3   0   2   -5   6
   |         
   |  3  

2. Multiply the value you just wrote down by the divisor root (1):

   • \( 3 \times 1 = 3 \)

3. Add this value to the next coefficient (0):

   • \( 0 + 3 = 3 \)

1 |  3   0   2   -5   6
   |        3
   |  3   3  

4. Repeat steps 2 and 3 for the remaining coefficients:
   • Multiply \( 3 \times 1 = 3 \), then add to 2:
     • \( 2 + 3 = 5 \)
   • Multiply \( 5 \times 1 = 5 \), then add to -5:
     • \( -5 + 5 = 0 \)
   • Multiply \( 0 \times 1 = 0 \), then add to 6:
     • \( 6 + 0 = 6 \)

1 |  3   0   2   -5   6
   |        3   5   0
   |  3   3   5   0   6

Step 4: Write down the result

The bottom row contains the coefficients of the quotient polynomial and the remainder.

The quotient polynomial is:

• Coefficients: \( 3, 3, 5, 0 \) corresponds to \( 3x^3 + 3x^2 + 5x + 0\)

The remainder is \( 6 \).

Final result

Thus, we can express the original division as: \[ 3x^4 + 2x^2 - 5x + 6 = (x - 1)(3x^3 + 3x^2 + 5x) + 6 \]

Quotient: \( 3x^3 + 3x^2 + 5x \)
Remainder: \( 6 \)