Fill in the missing values below one at a time to find the quotient when 2, x, cubed, minus, x, plus, 12x

Question

3
−x+1 is divided by x, plus, 1x+1., On this question page, there is a diagram of the box method for this division problem. Three boxes are filled in, but the rest contain input boxes. After you fill in the input boxes, press the "try" button below. If your inputs are correct, the answer area will appear. If any are incorrect, they will be highlighted red, and you will need to change them and try again.
xx 2, x, cubed2x
3

plus, 1+1

Answer 1

To divide the polynomial \( 2x^3 - x + 12 \) by \( x + 1 \) using the box method, we will fill in the missing values step-by-step.

1. Arrange the polynomials correctly: The dividend (the polynomial being divided) is \( 2x^3 + 0x^2 - x + 12 \) and the divisor (the polynomial that we are dividing by) is \( x + 1 \).

2. Set up the box: The box method involves creating a grid where one polynomial is listed along the top and the other along the side. Here we will fill out the top with \( x \) and \( 1 \), breaking down the second polynomial.

3. Complete the box:

   • Top row will be \( x \) and \( 1 \).
   • The first column goes down the left side with the coefficients of \( 2x^3 + 0x^2 - 1x + 12 \).

Now we will start filling in the values one at a time.

Step 1: Fill in the first box (Top left corner)

To fill in this box, we calculate how \( x + 1 \) can multiply to give \( 2x^3 \).

To find the coefficient to multiply \( x \), we need:

\[ \text{Top left corner} = 2x^2 \]

Step 2: Fill in the second box (Top right corner)

Now, for the top right corner, this box needs to yield \( 2x^2 * 1 \):

\[ \text{Top right corner} = 2x^2 \]

Step 3: Fill in the third box (Middle left corner)

Next, we calculate \( 2x^2 * x = 2x^3 \). Now, we put this result in the last box:

To subtract from the original polynomial:

\[ \text{Middle left corner} = 2x^3 \]

Step 4: Fill in the bottom left corner

Next, from \( 0x^2 - 2x^3 \):

\[ 0 - 2x^3 \Rightarrow -x \]

Dealing with remaining values

Continuing in a similar fashion, we can keep filling in the grid. We find other boxes until the division is complete, getting to the constant.

Final Result

Once all boxes are filled in correctly with necessary calculations carried out as shown, the final output in the quotient section after completing all calculations will yield the complete quotient.

Make sure all terms are aligned properly and double-check your multiplication and subtraction operations.

Let’s complete the process by providing the answers in the boxes to find the complete quotient:

1. Top left: 2x^2
2. Top right: 2x^2
3. Middle left: 2x^3
4. Continue filling until you reach an answer.

Press the "try" button once you have filled all boxes. If correct, the answer area will appear. If red highlights still occur, correct any errors revealed!