Question: 3 – Multiplying Monomials and Binomials
LEARNING OBJECTIVE: Multiply a monomial by a binomial.
What is the product of the $4x$ and the $3x^3y^2 - 2xy^3$?
a.) $12x^3y^2 - 2x^2y^3$
b.) $12x^4y^2 - 8x^2y^3$
c.) $12x^3y^2 - 8xy^3$
d.) $12x^4y^2 - 2xy^3$
1. **State the problem:** Multiply the monomial $4x$ by the binomial $3x^3y^2 - 2xy^3$.
2. **Recall the distributive property:** To multiply a monomial by a binomial, multiply the monomial by each term of the binomial separately and then add the results.
3. **Apply the distributive property:**
$$4x \times (3x^3y^2 - 2xy^3) = 4x \times 3x^3y^2 - 4x \times 2xy^3$$
4. **Multiply the coefficients:**
$$4 \times 3 = 12$$
$$4 \times 2 = 8$$
5. **Multiply the variables using the laws of exponents:**
- For $4x \times 3x^3y^2$: multiply $x$ and $x^3$ to get $x^{1+3} = x^4$, and keep $y^2$.
- For $4x \times 2xy^3$: multiply $x$ and $x$ to get $x^{1+1} = x^2$, and keep $y^3$.
6. **Write the product terms:**
$$12x^4y^2 - 8x^2y^3$$
7. **Final answer:** The product is $12x^4y^2 - 8x^2y^3$, which corresponds to option b.)