Final answer:
To divide (3x³ - x² - x - 1) by (x - 1), you can use polynomial long division. The quotient is 3x² + 2x + 1. Option C is correct.
Step-by-step explanation:
To divide (3x³ - x² - x - 1) by (x - 1), we can use polynomial long division.
First, divide the leading term of the dividend (3x³) by the leading term of the divisor (x) to get 3x².
Multiply the divisor (x - 1) by the quotient term (3x²) to get 3x³ - 3x².
Subtract this product from the dividend (3x³ - x² - x - 1) to get -2x² - x - 1.
Repeat the process by dividing the leading term of the new dividend (-2x²) by the leading term of the divisor (x) to get -2x.
Multiply the divisor (x - 1) by the new quotient term (-2x) to get -2x³ + 2x².
Subtract this product from the previous result (-2x² - x - 1) to get 3x² + 2x - 1.
There are no more terms to divide, so the final result is 3x² + 2x - 1.
Therefore, the quotient of (3x³ - x² - x - 1) divided by (x - 1) is 3x² + 2x - 1.
The question asks us to find the quotient of the division of two polynomials, (3x³ - x² - x - 1) divided by (x - 1). To find the quotient, we can use polynomial long division or synthetic division. In this case, synthetic division is quicker and more efficient.
Setting up synthetic division:
1 | 3 -1 -1 -1 (Coefficients of 3x³, -x², -x, and -1)
|_____
| 3 2 1 0 (New coefficients found by synthetic division)
When we apply synthetic division, we can determine the coefficients of the quotient polynomial, which gives us 3x² + 2x + 1. The remainder is 0, implying that (x - 1) is a factor of the original polynomial. Therefore, the correct answer is option c: 3x² + 2x + 1.