Problem 126 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. Find two factors of \(-35\) whose sum is \(2 .\)

Step-by-Step Solution

Answer

The two factors of -35 whose sum is 2 are -5 and 7.

Step 1: Identify the numbers to work with

Since the two numbers multiply to -35 and add up to 2, let's systematically find such numbers. The possible pairs of numbers getting on multiplication to 35 are (1,35), (-1,-35), (5,7), (-5,-7). Among these pairs only -5 and 7 add up to 2.

Step 2: Check the validity of the pair

Make sure if this pair of numbers is valid or not. Multiply to see if they give -35 and add to see if they give 2. Thus, \((-5) \times 7 = -35\) and \(-5+7=2\).

Step 3: Final Answer

Since both the criteria are satisfied by -5 and 7, these are the two factors of -35 whose sum is 2.

Key Concepts

Factorization Fundamentals

Factorization is the process of breaking down a number or expression into its constituent factors, which multiply together to produce the original number or expression.
It’s like peeling back layers to see what’s inside. Factorization is essential when solving equations because it helps simplify complex problems.In the exercise we're tackling, the goal is to factor the number \(-35\) into two numbers whose sum is \(2\). This means we want to find numbers that not only multiply to give us \(-35\) but also, when added together, equal \(2\).### Why Factorization Matters- **Simplification**: Makes equations easier to solve.- **Reveals properties**: Shows how numbers relate to each other.- **Foundation for advanced topics**: Useful for polynomials, quadratic equations, and more.Understanding factorization helps streamline problem-solving by reducing the complexity of equations.

Mastering Integer Operations

Working with integers involves understanding how to perform operations such as addition, subtraction, multiplication, and division with whole numbers. This foundation is crucial for tackling more advanced math problems.In our exercise, once we identify the numbers \(-5\) and \(7\), we perform integer operations to ensure they meet the criteria:### Integer Multiplication and Addition- **Multiplication**: When you multiply \(-5\) and \(7\), you get the product \(-35\). This confirms one part of the condition.- **Addition**: By adding \(-5\) and \(7\), their sum is \(2\), meeting the second condition.Remember:- **Signs matter**: Positive and negative signs affect the outcome significantly.- **Zero as a boundary**: Zero isn't positive or negative, but understanding its role helps in comprehending integers better.Grasping integer operations means being able to switch smoothly between adding, subtracting, multiplying, and dividing whole numbers.

Effective Problem-Solving Techniques

Problem-solving in math involves systematically working through a problem to find a solution. In our example, we use a logical approach to solve for two factors based on their sum and product.### Steps to Problem-Solving1. **Understand the Problem**: Clearly know what is being asked. Identify that you need factors of \(-35\) that add to \(2\).2. **Explore Possibilities**: Look at all factor pairs. For \(35\), consider \(1, 35\), \(5, 7\), and their negative equivalents.3. **Test and Verify**: Check each pair to see if they meet both criteria (product and sum).4. **Conclude and Reflect**: Confirm that \(-5\) and \(7\) satisfy the conditions.Using these techniques helps:- **Clarify the problem**: Cuts down on confusion by outlining steps.- **Logical thinking**: Encourages a structured approach.- **Accuracy checks**: Ensures the solution is correct through verification.By adopting these methods, you improve your math problem-solving abilities significantly.