Divide 2 x cubed minus 7 x squared plus 2 x plus 3 by x minus 3 using long division.

Question

(1 point)
Responses

2 x squared minus 13 x plus 41 R negative 120
Image with alt text: 2 x squared minus 13 x plus 41 R Image with alt text: negative 120

8 x squared plus 17 x plus 53 R 162
Image with alt text: 8 x squared plus 17 x plus 53 R 162

2 x cubed plus 3 x plus 1
Image with alt text: 2 x cubed plus 3 x plus 1

2 x squared minus x minus 1
Image with alt text: 2 x squared minus x minus 1

Answer 1

To perform long division of \( 2x^3 - 7x^2 + 2x + 3 \) by \( x - 3 \), follow these steps:

1. Divide the leading term of the numerator \( 2x^3 \) by the leading term of the denominator \( x \): \[ 2x^3 \div x = 2x^2 \]

2. Multiply \( 2x^2 \) by \( x - 3 \) and subtract from the original polynomial: \[ (2x^3 - 7x^2 + 2x + 3) - (2x^2 \cdot (x - 3)) = (2x^3 - 7x^2 + 2x + 3) - (2x^3 - 6x^2) = -1x^2 + 2x + 3 \]

3. The next term is \( -1x^2 \). Divide by \( x \): \[ -1x^2 \div x = -1x \]

4. Multiply \( -1x \) by \( x - 3 \) and subtract: \[ (-x^2 + 2x + 3) - (-1x \cdot (x - 3)) = (-x^2 + 2x + 3) - (-x^2 + 3x) = -x + 3 \]

5. Divide \( -x \) by \( x \): \[ -x \div x = -1 \]

6. Multiply \( -1 \) by \( x - 3 \) and subtract: \[ (-x + 3) - (-1 \cdot (x - 3)) = (-x + 3) - (-x + 3) = 0 \]

So, there is no remainder, and the result from the division is: \[ 2x^2 - x - 1 \]

Thus, the answer is: 2 x squared minus x minus 1