A student used synthetic division to divide x^3-3x^2+4 by x+1 .
Mark and explain where the error(s) occurred in the student’s setup for synthetic division. (2 points)
After marking the errors, correctly perform synthetic division. (2 points)
Using the Remainder Theorem (p. 307), is your answer and x+1. a factor of the polynomial ? If so, explain how you know. (1 point)
If your answer to part c is yes, use the Factor Theorem (p.290) to show the fully factored form of the polynomial x^3-3x^2+4 . (1 point)
The errors in the student's setup for synthetic division are:
1. The student forgot to include the coefficient of x^2 in the division process.
2. The student wrote the coefficients of x^2, x, and the constant term incorrectly.
To perform synthetic division correctly, we need to include all the terms and write the coefficients correctly. So the setup should be:
-1 | 1 -3 0 4
Performing synthetic division:
-1 | 1 -3 0 4
| -1 4 -4
-------------
1 -4 4 0
Using the Remainder Theorem, the remainder is 0. Since x+1 divides evenly into the polynomial, it is a factor of the polynomial. We know this because when x=-1, the polynomial evaluates to 0.
Using the Factor Theorem, if x+1 is a factor of the polynomial, then (x+1) must be one of the factors in the fully factored form. By dividing x^3 - 3x^2 + 4 by x+1, we obtained the quotient 1x^2 - 4x + 4. So the fully factored form of the polynomial x^3 - 3x^2 + 4 is (x+1)(x^2 - 4x + 4).