In Exercises 1–68, factor completely, or state that the polynomial is prime. x + 8x⁴

Verified step by step guidance

1. Step 1: Identify the common factor in the polynomial terms.
2. Step 2: Notice that both terms, x and 8x^4, have a common factor of x.
3. Step 3: Factor out the common factor x from the polynomial.
4. Step 4: Write the expression as x(1 + 8x^3).
5. Step 5: Check if the expression inside the parentheses, 1 + 8x^3, can be factored further.

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying methods such as grouping or the quadratic formula for higher-degree polynomials.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, identifying the GCF allows for simplification by factoring it out, which can make the remaining polynomial easier to work with. For example, in the expression x + 8x⁴, the GCF is x, which can be factored out.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is crucial in algebra, as it indicates that the polynomial cannot be simplified further. In the case of x + 8x⁴, after factoring out the GCF, the remaining polynomial must be assessed to determine if it can be factored further or if it is prime.