Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 5(3 x-7)+x(3 x-7) $$

Short answer

The polynomial can be factored as \[ (3x - 7)(5 + x) \].

Step-by-step solution

Identify the common factor

Observe the polynomial to find any common factors. Notice that both terms in the polynomial share a common factor of \(3x - 7\).

Factor out the common factor

Factor out the common factor \(3x - 7\). This means the original expression can be rewritten as: \[ (3x - 7)(5 + x) \]

Key concepts

Common Factor

When working with polynomials, identifying a common factor can make the factoring process simpler. A common factor is a term that appears in each part (or term) of the polynomial. It's like finding the greatest common divisor but for expressions. In our example, the polynomial is $$ 5(3x - 7) + x(3x - 7) $$. By looking at this, you can see that both parts share the term (3x - 7\text).
This term appears in both the first term, (5(3x - 7)\text), and the second term, (x(3x - 7)\text). Identifying and factoring out this common term will help simplify the expression and lead us to the next steps.It's crucial to spot these shared factors as it can greatly streamline the factoring process.

Factoring Process

Factoring a polynomial requires several steps, and it always starts with identifying any common factors. Once you have identified a common factor, you can factor it out and simplify the polynomial. In our example, the common factor is (3x - 7\text).By factoring out (3x - 7\text), we take that term out of each part of the polynomial and place it in front of a set of parentheses.
The original expression changes from $$( 5(3x - 7) + x(3x - 7)\text) to (3x - 7)(5 + x\text).
Now, instead of dealing with a polynomial with multiple terms, you have a simpler product of two factors: (3x - 7)\text) and (5 + x\text). This simplification makes it easier to understand and work with the expression. Always remember, after factoring, to double-check your work by distributing to ensure you get back to the original expression.

Identifying Prime Polynomials

Sometimes, a polynomial cannot be factored further, and we call these 'prime polynomials'. A prime polynomial is one that does not have any factors other than 1 and itself. For example, consider the polynomial (3x + 2\text). There are no common factors in each term, nor can it be simplified to a product of other simpler polynomials. Therefore, it is considered prime.
In our example, $$(3x - 7)(5 + x)\text), neither (3x - 7\text) nor (5 + x\text) can be factored further. To determine if a polynomial is prime:

• Look for any common factors within the terms.
• Try to rewrite the polynomial as a product of other polynomials.
• If you cannot find such factors or rewrite it further, it is likely prime.

Identifying prime polynomials helps conclude the factoring process, ensuring you have simplified the expression as much as possible.