Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 4-14 x^{2}-8 x^{4} $$

Short answer

The polynomial -8x^4 - 14x^2 + 4 is prime.

Step-by-step solution

- Arrange the Polynomial

Arrange the polynomial in descending order of powers of x. The expression becomes -8x^4 - 14x^2 + 4.

- Identify the Greatest Common Factor (GCF)

Factor out any common factors from all the terms. In this case, there is no common factor among -8, -14, and 4.

- Rearrange the Polynomial

Rewrite the polynomial in a more manageable form: -8x^4 - 14x^2 + 4.

- Check if it Can Be Factored

To determine if the polynomial can be factored further, try and factor by using techniques such as grouping or the quadratic form. For this polynomial, these techniques do not apply, and it cannot be simplified further.

- Conclusion

Since no further factoring is possible, the polynomial is considered a prime polynomial.

Key concepts

polynomial factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. Factors are simpler polynomials that multiply together to give the original polynomial. Factorization is a key tool in simplifying expressions and solving polynomial equations. Common methods for factorizing polynomials include:

• Factoring out the greatest common factor (GCF)
• Grouping terms
• Using special formulas (like the difference of squares)
• Quadratic factorization

Polynomials can only be factored if they can be expressed as a product of simpler polynomials. If not, they are considered prime.

greatest common factor

The greatest common factor, or GCF, of a set of terms is the highest number and variable power that divides each term in the set without leaving a remainder. For example, in the polynomial \(6x^3 + 9x^2 - 12x\), the GCF of the terms is 3x. Factoring out the GCF simplifies polynomials and is often the first step in the factorization process. To find the GCF:

• List the factors of each coefficient
• Identify the common factors
• Choose the highest power of x common to all terms

Factoring out the GCF transforms the polynomial into a simpler form, making it easier to work with in subsequent steps.

prime polynomial

A prime polynomial is a polynomial that cannot be factored into the product of two or more non-constant polynomials. Just as prime numbers have no divisors other than 1 and themselves, prime polynomials can only be divided by 1 and themselves. Identifying a prime polynomial involves checking if it can be factorized using various methods such as:

• Factoring out the GCF
• Using special formulas
• Quadratic formula

If none of these methods work and no suitable factors are found, the polynomial is considered prime. For example, the polynomial \(x^2 + x + 1\) has no factors, making it prime.

quadratic form

A polynomial is in a quadratic form when it can be expressed in the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Even higher-degree polynomials can sometimes be written in a quadratic-like form. The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is often used to find the roots of quadratic polynomials. Determining if a polynomial can be expressed in a quadratic form can simplify solving and factoring it. This involves re-writing and rearranging terms to fit the quadratic structure, which is a critical step in solving polynomial equations.