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. **Formula and rules:** To multiply a monomial by a binomial, use the distributive property:
$$a(b + c) = ab + ac$$
Here, multiply $4x$ by each term inside the parentheses.
3. **Multiply the first term:**
$$4x \times 3x^3y^2 = 4 \times 3 \times x \times x^3 \times y^2 = 12x^{1+3}y^2 = 12x^4y^2$$
4. **Multiply the second term:**
$$4x \times (-2xy^3) = 4 \times (-2) \times x \times x \times y^3 = -8x^{1+1}y^3 = -8x^2y^3$$
5. **Combine the results:**
$$12x^4y^2 - 8x^2y^3$$
6. **Check answer choices:** This matches option b.)
**Final answer:** $12x^4y^2 - 8x^2y^3$ (option b.)