Problem 17 - Elementary Algebra

Question

Factor completely. $$ 2 \times 5-162 \times 3 $$

Step-by-Step Solution

Answer

The expression simplifies and factors to \(-1 \times 2^2 \times 7 \times 17\).

Step 1: Identify Terms

The expression given is \(2 \times 5 - 162 \times 3\). Here, we have two terms to consider: \(2 \times 5\) and \(-162 \times 3\). First, we will calculate each product separately.

Step 2: Simplify Each Term

Calculate the product of each term:\[2 \times 5 = 10\]\[162 \times 3 = 486\]So, the expression simplifies to:\[10 - 486\]

Step 3: Combine Like Terms

Subtract the two simplified expressions:\[10 - 486 = -476\]Now, the expression is \(-476\).

Step 4: Factor the Result

We need to factor \(-476\). First, find the factors of 476. It is divisible by 2 (since it's even):\[476 \div 2 = 238\]Next, factor 238, which is also divisible by 2:\[238 \div 2 = 119\]Now, factor 119, which is divisible by 7:\[119 \div 7 = 17\]Finally, since 17 is a prime number, the factors of 476 are 2, 2, 7, and 17. Therefore, \(-476\) factors to:\[-(2^2 \times 7 \times 17)\]Therefore, the complete factorization, considering the negative sign, is:\[-1 \times 2^2 \times 7 \times 17\]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 with no divisors other than 1 and itself. These are the nuggets of numbers because they can

• only be divided evenly by 1 and themselves.
• act as the fundamental stepping stones for creating any larger number.

To factor a number into its prime components,

• start by dividing the number by the smallest prime, generally 2, and continue with other primes (3, 5, 7, etc.) until you can't divide anymore.

In our exercise, the number 476 was broken down into prime factors by sequentially dividing it by 2 until 119, then 7, and finally reaching 17, a prime number. This gave us the prime factorization of 476 as \(2^2 \times 7 \times 17\). With negative numbers included, as is the case with -476, the negative sign is simply accounted for separately, but does not affect the prime computation.

Simplification

Simplification in algebra involves reducing an expression to its most concise form. This means streamlining terms and numbers to make them as straightforward as possible. In our example, the term-by-term multiplication led to:

• First term: \(2\times5\) which results in 10.
• Second term: \(162\times3\) which simplifies to 486.

After you simplify each term, it's essential to combine them, as seen with \(10 - 486\). The result, \(-476\), is now the simplest form of the original problem. The goal of simplification is to make calculations easier, eliminate potential errors from complex expressions, and provide an approachable result.

Negative Numbers

Negative numbers represent quantities less than zero and have unique properties in math. They are crucial in maintaining balance in equations by:

• indicating things like debts or temperatures below zero.
• being located to the left of zero on the number line.

When dealing with expressions involving negatives, keep these points in mind:

• Subtracting a larger number from a smaller one results in a negative answer, as seen in \(10 - 486 = -476\).
• When factoring, negative numbers influence the sign of the final result. For instance, factoring \(-476\) introduces a \(-1\) multiplied across the prime factorization, resulting in \(-1 \times 2^2 \times 7 \times 17\).

Embracing negative values in algebra helps solve a wide range of problems, from financial calculations to scientific measurements.