Problem 11 - Elementary Algebra

Question

Factor completely. $$ 144 \times 2-1 $$

Step-by-Step Solution

Answer

The expression simplifies to 287, which is a prime number and cannot be factored further.

Step 1: Simplify the Expression

Start by simplifying the expression. Multiply 144 by 2 to get 288. Therefore, the expression becomes \( 288 - 1 \).

Step 2: Calculate the Difference

Subtract 1 from 288 to simplify the expression further. This results in \( 287 \).

Step 3: Verify if the Result Can Be Factored

Check if the number 287 can be factored further by testing if it is a prime number. Verify potential divisibility by smaller prime numbers like 2, 3, 5, 7, and 11. None of these divide 287 perfectly. Thus, 287 is a prime number.

Key Concepts

Factoring

Factoring is the process of breaking down a mathematical expression into simpler components that can be multiplied to produce the original expression. In this exercise, we started with an expression: \(144 \times 2 - 1\). The goal is to determine if the result can be factored further. To factor successfully, follow these steps:

• Look for a common factor in the terms of your expression, if there are multiple terms.
• If the result is a single number, see if it can be split into a product of other numbers.

In this case, after simplifying the expression, we obtained 287. Therefore, our task is to determine if 287 can be broken down into smaller numbers that multiply to give 287, which leads us to the next steps involving simplification and prime testing.

Simplification

Simplification involves performing basic arithmetic operations to reduce an expression to its most straightforward form. Our initial expression was \(144 \times 2 - 1\).Here's how we simplified it:

• Multiply \(144\) by \(2\) to get \(288\).
• Subtract \(1\) from \(288\) to arrive at \(287\).

After simplification, our original expression was reduced to the number 287. This process not only makes it easier to work with numbers, but it also prepares the way for further factorization or simplification steps, such as checking for primality.

Prime Number Verification

Prime number verification is the process of determining whether a given number is a prime. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To verify if 287 is a prime number:

• Check if it is divisible by small prime numbers such as 2, 3, 5, 7, and 11.
• If none of these numbers divide 287 perfectly (i.e., no remainder), 287 is a prime number.

For 287, testing divisibility by these primes showed that none of them divide 287 without a remainder. Thus, we confirm that 287 is indeed a prime number, meaning it cannot be factored further. Understanding this process is crucial for recognizing prime numbers and simplifying expressions efficiently.