To use synthetic division to simplify the expression x^4-x^3-3x-2 divided by (x-2), we follow these steps:
1. Write the coefficients of the polynomial in descending order, leaving a placeholder for any missing terms. In this case, we have:
1 | 1 -1 -3 0 -2
where the first term, 1, represents the coefficient of x^4, the second term, -1, represents the coefficient of x^3, and so on.
2. Write the constant term of the divisor (x-2) outside the division bracket and draw a line underneath it. In this case, the constant term is -2, so we have:
-2 | 1 -1 -3 0 -2
|
3. Bring down the first coefficient, which is 1, and write it underneath the line. This represents the value of the polynomial when x = 0.
-2 | 1 -1 -3 0 -2
| 1
4. Multiply the constant term outside the bracket by the first term inside the bracket and write the result underneath the second coefficient of the dividend. In this case, we have:
-2 | 1 -1 -3 0 -2
| 1 6
_________
1 0 3 0 -2
5. Repeat the process by bringing down the next coefficient, which is 0, and placing it underneath the line. Then, multiply the constant term by the second term of the dividend and write the result underneath the third coefficient. Continue this process until all the coefficients have been brought down and multiplied.
-2 | 1 -1 -3 0 -2
| 1 6 -6
_________
1 0 3 6 4
6. The final row of numbers represents the coefficients of the simplified polynomial, read from left to right. Therefore, x^3 + 3x^2 + 6x + 4 is the simplified form of x^4-x^3-3x-2 divided by (x-2).
So, the answer is:
x^4-x^3-3x-2 = (x-2)(x^3 + 3x^2 + 6x + 4)