Question

the volume of a rectangular prism is (2x^3 + 9x^2 - 8x - 38) with height (x + 2). using synthetic division, what is the area of the base?

(a) (2x^3 + 13x^2 + 18x)
(b) (2x^3 + 5x^2 - 18x)
(c) (2x^2 + 13x + 18)
(d) (2x^2 + 5x - 18)

Explanation:

🆕 New Concept Discovered: Polynomial Division via Synthetic Division
A shorthand method to divide polynomials by linear factors

Step 1: Set up the division

The volume of a rectangular prism is given by:

$$ V = \text{Area of the base} \times \text{height} $$

To find the area of the base, we divide the volume polynomial by the height:

$$ \text{Area} = \frac{2x^3 + 9x^2 - 8x - 36}{x + 2} $$

For synthetic division with a divisor of \( x + 2 \), we use the root \( c = -2 \) and write down the coefficients of the dividend:

$$ 2 \quad 9 \quad -8 \quad -36 $$

Step 2: Perform synthetic division

Bring down the first coefficient, multiply by \( -2 \), add to the next coefficient, and repeat:

 -2 |   2    9   -8   -36
    |       -4  -10    36
    ---------------------
        2    5  -18     0

Step 3: Write the resulting polynomial

The numbers at the bottom represent the coefficients of the quotient polynomial, which is one degree lower than the original cubic polynomial:

$$ 2x^2 + 5x - 18 $$

Answer:

\( 2x^2 + 5x - 18 \)