To perform synthetic division for the polynomial \(2x^3 - 2x^2 + 6x - 4\) by \(x - 2\), we will use 2 (the zero of \(x - 2\)) in the synthetic division process. Here's how to set it up and perform the division:
-
Write down the coefficients of the polynomial. For \(2x^3 - 2x^2 + 6x - 4\), the coefficients are:
- \(2\) (for \(2x^3\))
- \(-2\) (for \(-2x^2\))
- \(6\) (for \(6x\))
- \(-4\) (for the constant term)
-
Set up the synthetic division:
2 | 2 -2 6 -4 |_________________ -
Bring down the leading coefficient (2):
2 | 2 -2 6 -4 |_________________ | 2 -
Multiply the value you've just brought down (2) by 2 (the number outside) and write the result under the next coefficient:
2 | 2 -2 6 -4 | 4 |_________________ | 2 2 -
Add the column: \(-2 + 4 = 2\)
2 | 2 -2 6 -4 | 4 4 |_________________ | 2 2 10 -
Repeat the multiplication and addition for the next column:
- Multiply \(2\) (the last result) by \(2\) (the number outside): \(2 \times 2 = 4\)
- Add: \(6 + 4 = 10\)
2 | 2 -2 6 -4 | 4 4 20 |_________________ | 2 2 10 -
Finally, multiply \(10\) by \(2\): \(10 \times 2 = 20\) and add: \(-4 + 20 = 16\)
2 | 2 -2 6 -4 | 4 4 20 |_________________ | 2 2 10 16
The final row (the bottom row) gives us the coefficients of the quotient and the remainder:
- The quotient is \(2x^2 + 2x + 10\)
- The remainder is \(16\)
Thus, the result of \(2x^3 - 2x^2 + 6x - 4\) divided by \(x - 2\) using synthetic division is:
\[ 2x^3 - 2x^2 + 6x - 4 = (x - 2)(2x^2 + 2x + 10) + 16 \]
1 answer