If the polynomial x^3+6x^2+11x+6 expresses the volume, in cubic inches, of the box, and the width is (x+1)in., what are the dimensions of the box?

I need to answer the question in long dvision or synthetic division, the thing is i have no idea how to do either of those.
quick explanation?

a little synthetic division quickly shows that

x^3+6x^2+11x+6 = (x+1)(x^2+5x+6) = (x+1)(x+2)(x+3)

Am i capable of taking this question down? no one answered and ive gotten the information i needed from other sources.

Long division and synthetic division are two methods used to divide polynomials. In this case, we need to find the dimensions of the box, given the polynomial expression for its volume.

Let's start with long division:
1. Write the polynomial expression for the volume and divide it by (x+1), the expression for the width of the box.
x^2 + 5x + 6 | x^3 + 6x^2 + 11x + 6

2. Compare the leading term of the divisor (x+1) to the leading term of the dividend (x^3), and divide x^3 by x, which gives x^2.
x^2

3. Multiply the x^2 by the divisor (x+1) and subtract the result from the dividend.
- (x^3 + x^2)

4. Bring down the next term from the dividend (- x^3 - x^2 + 6x^2).
+ 6x^2 + 11x + 6

5. Compare the leading term of the new dividend (6x^2) to the leading term of the divisor (x+1) and divide 6x^2 by x, which gives 6x.
x^2 + 6x

6. Multiply the 6x by the divisor (x+1) and subtract the result from the new dividend.
- (6x^2 + 6x)

7. Bring down the next term from the new dividend (- 6x^2 - 6x + 11x).
+ 5x + 6

8. Continue comparing and dividing until there are no more terms in the dividend.
- (5x^2 + 5x)

9. Bring down the constant term from the new dividend (- 5x^2 - 5x + 5x).
+ 6

10. There are no more terms to divide, so the quotient is x^2 + 6x + 5.

The dimensions of the box are x+1 inches, x^2 + 6x + 5 inches, and x^3 + 6x^2 + 11x + 6 inches.

If you still have difficulty understanding this method, I can explain synthetic division as an alternative.

To find the dimensions of the box given the polynomial, you can use either long division or synthetic division. Let me explain both methods briefly:

1. Long Division:
- Write the polynomial in standard form: x^3 + 6x^2 + 11x + 6.
- Set up the long division, dividing by (x+1), similar to dividing numbers.
- Start dividing the terms.
- The quotient will be a polynomial that represents the length and height of the box, and the remainder will represent the depth.

2. Synthetic Division:
- Write down the coefficients of the polynomial (1, 6, 11, 6).
- Reverse the sign of the constant term (6) and bring it down.
- Multiply this term by the divisor's root (-1) and write the result underneath the next coefficient.
- Add the column of numbers.
- Repeat the process until you reach the end.
- The numbers in the last row represent the coefficients of the quotient polynomial.

Both methods will give you the same result, but synthetic division is a shortcut and requires less writing. Here is the step-by-step procedure.

Using Synthetic Division:
1. Write down the coefficients: 1, 6, 11, 6.
2. Set up the division like this, but only include the coefficients:

-1 | 1 6 11 6

3. Bring down the first coefficient (1).
4. Multiply -1 by 1, and write the result under the next coefficient:

-1 | 1 6 11 6
-1

5. Add the numbers in the first column:

-1 | 1 6 11 6
-1
_________________

-1 5

6. Multiply -1 by 5, and write the result under the next coefficient:

-1 | 1 6 11 6
-1 -5

7. Add the numbers in the second column:

-1 | 1 6 11 6
-1 -5
_________________

-1 5 6

The resulting numbers in the last row are the coefficients of the quotient polynomial. In this case, the quotient is x^2 + 5x + 6.

Therefore, the dimensions of the box are:
- Length (L) = x + 1
- Height (H) = x^2 + 5x + 6
- Depth (D) = -1

So, the dimensions of the box are (L, H, D) = (x+1, x^2+5x+6, -1).