Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 14+6 x-x^{2} $$

Short answer

Polynomial is prime: -1(x^2 - 6x - 14).

Step-by-step solution

Reorder the terms

First, reorder the terms of the polynomial in descending powers of x. The given polynomial is 14 + 6x - x^2. Rearrange it to get: -x^2 + 6x + 14.

Factor out the common factor

Notice that the leading term has a negative coefficient. Factor out -1 from the polynomial: -1(x^2 - 6x - 14)

Factor the quadratic expression

Now, attempt to factor the quadratic expression inside the parentheses, x^2 - 6x - 14. Look for two numbers that multiply to -14 and add to -6. However, as no such pair of integers exists, conclude that the quadratic part is prime.

Conclusion

Since the expression inside the parentheses cannot be factored further, the original polynomial is -1(x^2 - 6x - 14). Therefore, the polynomial is considered to be prime as it does not factorize completely over the set of integers.

Key concepts

quadratic expressions

A quadratic expression is a type of polynomial with the highest power of the variable being 2. Generally, it is of the form \(ax^2 + bx + c\). Knowing how to manage quadratic expressions is essential since they appear frequently in math problems.
To solve or factor a quadratic expression, the standard form is useful: \(ax^2 + bx + c\). Try to identify possible methods like factoring, using the quadratic formula, or completing the square.
In our exercise, the quadratic expression is \(x^2 - 6x - 14\). Keep in mind to always reorder terms in descending power of x (if necessary), before attempting to factorize them.

prime polynomials

A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients. Just like prime numbers, prime polynomials are indivisible by other polynomials.
In the exercise provided, after attempting to factor \(x^2 - 6x - 14\), no two numbers multiply to -14 and add to -6. Hence, we conclude that this quadratic expression is a prime polynomial.
Always verify if further decomposition isn't possible. Double-check your calculations when concluding a polynomial is prime.

factoring techniques

Factoring techniques are essential tools for simplifying polynomials. Here are a few common ones:

• Factoring out the greatest common factor (GCF): Always start by checking for a GCF in all terms.
• Grouping: Used for polynomials with four or more terms.
• Factoring trinomials: Find two numbers that multiply to \(ac\) and add to \(b\) in \(ax^2 + bx + c\).
• Difference of squares: Recognize expressions of the form \(a^2 - b^2 = (a+b)(a-b)\).

In the exercise provided, first, reorder the terms. Then, by factoring out -1, an attempt to factor the quadratic trinomial \( -x^2 + 6x + 14\) reveals that it is prime, something only concluded after trying to apply different techniques.