how do u do synthetic division with the problem:
(x^5-7x^3+x+1)/ x+3
To perform synthetic division, follow these steps:
Step 1: Write the problem and identify the divisor.
Dividend (Polynomial): x^5 - 7x^3 + x + 1
Divisor: x + 3
Step 2: Set up the synthetic division table.
First, write down the coefficients of the dividend, in decreasing order of their exponents. Fill in any missing terms with zeroes. In this case, the dividend's missing terms are x^4 and x^2, which have coefficients of zero.
Expanding the dividend: x^5 - 7x^3 + 0x^4 + 0x^2 + x + 1
Write the coefficients: 1, 0, -7, 0, 1, 1
Next, write the divisor on the left side of the table. In this example, the divisor is x + 3.
x + 3 |
Step 3: Perform the synthetic division.
Start by bringing down the first coefficient, which is 1, below the horizontal line.
_______________________
x + 3 | 1 0 -7 0 1 1
Next, multiply the divisor (x + 3) by the first term (1) and write the result below the next coefficient, which is 0. The result is then added to the coefficient below it.
_______________________
x + 3 | 1 0 -7 0 1 1
3
_______________________
1 3
Repeat the process by multiplying the divisor (x + 3) by the new term (3). Write the result below the next coefficient, which is -7. Again, the result is added to the coefficient below it.
_______________________
x + 3 | 1 0 -7 0 1 1
3 12
_______________________
1 3 5
Repeat these steps until you reach the last term.
_______________________
x + 3 | 1 0 -7 0 1 1
3 12 15
_______________________
1 3 5 15 16
Step 4: Write the result and interpret it.
The final row of the synthetic division table gives us the coefficients of the resulting polynomial after dividing. In this example, the result is 1x^4 + 3x^3 + 5x^2 + 15x + 16.
Therefore, the quotient is: x^4 + 3x^3 + 5x^2 + 15x + 16, and the remainder is 16.
To summarize:
(x^5 - 7x^3 + x + 1) ÷ (x + 3) = x^4 + 3x^3 + 5x^2 + 15x + 16 + 16/(x + 3)