Final answer:
The errors in the student's setup for synthetic division are misplacing the sign of the constant term and not including the coefficient of the x term. The correct setup for synthetic division is provided along with the steps. The Remainder Theorem and the Factor theorem are explained to determine if x+1 is a factor of the polynomial and the fully factored form of the polynomial is explained if it is a factor.
Step-by-step explanation:
The errors in the student's setup for synthetic division are:
- Misplacing the sign of the constant term.
- Not including the coefficient of the x term.
The correct setup for synthetic division is as follows:
x^3-x^2-2x / (x+1)
- Arrange the terms in descending order of degree.
- Write down the coefficients of each term, including the missing coefficients. In this case, the coefficient of x is 0.
- Reverse the sign of the constant term.
- Bring down the leading coefficient.
- Multiply the divisor by the first term.
- Subtract the result from the second term.
- Multiply the divisor by the new result.
- Repeat steps 6 and 7 until all terms have been divided.
- The resulting coefficients are the coefficients of the quotient polynomial.
Using the Remainder Theorem and the Factor theorem, we can determine if x+1 is a factor of the polynomial:
Remainder Theorem: If x+1 is a factor of the polynomial, the remainder when dividing the polynomial by x+1 will be 0.
Factor Theorem: If x+1 is a factor of the polynomial, then x = -1 is a solution to the polynomial equation.
To check if x+1 is a factor, we substitute x = -1 into the polynomial and see if it equals 0. If it does, then x+1 is a factor of the polynomial.
If x+1 is a factor, then the fully factored form of the polynomial can be found by dividing the original polynomial by x+1.