(3x-5y)(4x+7y)

Write the numerical coefficient.

Blank 1x2 + Blank 2xy + Blank 3y2

To find the numerical coefficients for each term in the expression \((3x - 5y)(4x + 7y)\), we first need to expand the product using the distributive property (also known as the FOIL method for binomials).

### Step-by-Step Expansion

1. **First terms:** \(3x \cdot 4x = 12x^2\)
2. **Outer terms:** \(3x \cdot 7y = 21xy\)
3. **Inner terms:** \(-5y \cdot 4x = -20xy\)
4. **Last terms:** \(-5y \cdot 7y = -35y^2\)

Now, combine like terms:

\[
12x^2 + (21xy - 20xy) - 35y^2
\]

Simplify the expression:

\[
12x^2 + 1xy - 35y^2
\]

### Numerical Coefficients

Now, we can identify the numerical coefficients for each term:

1. **Coefficient of \(x^2\):** \(12\)
2. **Coefficient of \(xy\):** \(1\)
3. **Coefficient of \(y^2\):** \(-35\)

Thus, the numerical coefficients are:
- **Blank 1 (for \(x^2\)):** 12
- **Blank 2 (for \(xy\)):** 1
- **Blank 3 (for \(y^2\)):** \(-35\)

Would you like more details or have any questions?

### Related Questions:

1. How do you multiply two binomials using the FOIL method?
2. What are like terms, and why do we combine them when simplifying expressions?
3. How do you find the numerical coefficient of a variable in an algebraic expression?
4. Can you provide an example of expanding and simplifying a different set of binomials?
5. What happens if one of the terms in a binomial expansion is negative?

**Tip:** When expanding binomials, always ensure to multiply each term in the first binomial by each term in the second binomial to avoid missing any terms.

Learn how to find the numerical coefficients of the expression (3x - 5y)(4x + 7y) using the FOIL method. Understand the steps involved in expanding and simplifying binomial expressions. Suitable for high school students studying algebra and polynomial operations.

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