A student used synthetic division to divide x3-3x2+4 by x+1 .
a. Mark and explain where the error(s) occurred in the student’s setup for synthetic division. (2 points)
b. After marking the errors, correctly perform synthetic division. (2 points)
c. Using the Remainder Theorem (p. 307), is your answer and .x+1 a factor of the polynomial x3-3x2+4 ? If so, explain how you know x3-3x2+4 (1 point)
d. If your answer to part c is yes, use the Factor Theorem (p.290) to show the fully factored form of the polynomial x3-3x2+4(1 point)
correct this answer 1[1-3+4 - 1-2+2
a. The error in the student's setup for synthetic division is that they did not include the correct signs for the coefficients of the polynomial. They wrote x^3 - 3x^2 + 4 instead of x^3 - 3x^2 + 0x + 4.
b. To correctly perform synthetic division, we set up the division as follows:
-1 | 1 -3 0 4
1 -1 1
___________________
1 -4 1 5
Therefore, the corrected synthetic division for x^3 - 3x^2 + 4 ÷ (x+1) is x^2 -4x + 1, with a remainder of 5.
c. According to the Remainder Theorem, if a polynomial function f(x) is divided by x-a, then the remainder r is equal to f(a). In this case, we divided x^3 - 3x^2 + 4 by x+1 and obtained a remainder of 5. Therefore, f(-1) = 5, which means that -1 is NOT a root of the polynomial x^3 - 3x^2 + 4. Hence, x+1 is NOT a factor of the polynomial x^3 - 3x^2 + 4.
d. Since x+1 is not a factor, we cannot use the Factor Theorem to fully factor the polynomial x^3 - 3x^2 + 4.