Questions: What is the product of the 2 x^3 and the 4 x^2 y-3 x y^2 ? a.) 8 x^5 y-6 x^4 y^2 b.) 8 x^5 y-3 x y^2 c.) 8 x^6 y-6 x^3 y^2 d.) 8 x^6 y-3 x y^2

Transcript text: What is the product of the $2 x^{3}$ and the $4 x^{2} y-3 x y^{2} ?$ a.) $8 x^{5} y-6 x^{4} y^{2}$ b.) $8 x^{5} y-3 x y^{2}$ c.) $8 x^{6} y-6 x^{3} y^{2}$ d.) $8 x^{6} y-3 x y^{2}$

Solution

Solution Steps

To find the product of a monomial and a binomial, we need to distribute the monomial to each term in the binomial. Specifically, we will multiply $2x^3$ by each term in the binomial $4x^2y - 3xy^2$.

Step 1: Define the Monomial and Binomial

We start with the monomial $2x^3$ and the binomial $4x^2y - 3xy^2$.

Step 2: Distribute the Monomial

We will distribute the monomial $2x^3$ to each term in the binomial: \[ 2x^3 \cdot (4x^2y) \quad \text{and} \quad 2x^3 \cdot (-3xy^2) \]

Step 3: Calculate Each Product

Calculating the first term: \[ 2x^3 \cdot 4x^2y = 8x^{5}y \] Calculating the second term: \[ 2x^3 \cdot (-3xy^2) = -6x^{4}y^{2} \]

Step 4: Combine the Results

Now, we combine the results from the two products: \[ 8x^{5}y - 6x^{4}y^{2} \]