Exercises 73–75 will help you prepare for the material covered in the next section. Multiply: (7 - 3x)(- 2 - 5x)

Verified step by step guidance

1. Identify the two binomials to be multiplied: \((7 - 3x)\) and \((-2 - 5x)\).
2. Apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial by each term in the second binomial: multiply \(7\) by \(-2\), then \(7\) by \(-5x\), then \(-3x\) by \(-2\), and finally \(-3x\) by \(-5x\).
3. Write out each product explicitly: \(7 \times (-2)\), \(7 \times (-5x)\), \(-3x \times (-2)\), and \(-3x \times (-5x)\).
4. Simplify each product by performing the multiplication and combining constants and variables appropriately.
5. Combine all the simplified terms into a single expression and then combine like terms if any exist.

Key Concepts

Distributive Property

The distributive property allows you to multiply each term inside one parenthesis by each term inside the other. It is essential for expanding expressions like (7 - 3x)(-2 - 5x) by multiplying every term in the first binomial by every term in the second.

Combining Like Terms

After expanding the expression, you often get multiple terms with the same variable part. Combining like terms means adding or subtracting these terms to simplify the expression into a standard polynomial form.

Polynomial Multiplication

Multiplying two binomials results in a polynomial, typically of degree two. Understanding how to multiply polynomials helps in simplifying expressions and solving equations involving polynomial terms.