Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{8}-x^{5} $$

Short answer

The polynomial is factored as: \( x^5 (x - 1)(x^2 + x + 1) \).

Step-by-step solution

Identify the Greatest Common Factor (GCF)

Look at both terms in the polynomial to find the greatest common factor. Here, both terms share a common factor of \(x^5\). So, we factor out \(x^5\) from the polynomial: \[ x^8 - x^5 = x^5 (x^3 - 1) \]

Factor the Remaining Polynomial

Now focus on the polynomial inside the parentheses: \(x^3 - 1\). Recognize that this is a difference of cubes, which can be factored using the formula for factoring a difference of cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] With \(a = x\) and \(b = 1\), we get: \[ x^3 - 1 = (x - 1)(x^2 + x + 1) \]

Combine All Factors

Now combine the factors found in previous steps. The completely factored form of the polynomial is: \[ x^8 - x^5 = x^5 (x - 1)(x^2 + x + 1) \]

Key concepts

Greatest Common Factor (GCF)

In algebra, finding the Greatest Common Factor (GCF) is a fundamental step when you're tasked with factoring polynomials. The GCF is the highest number or variable that evenly divides all terms in the given polynomial.

For the exercise above, the given polynomial is \(x^8 - x^5\). The first step involves identifying any common factors shared by each term.

We look at the terms \(x^8\) and \(x^5\). Notice that both terms include a factor of \(x^5\). Hence, the GCF for these terms is \(x^5\).

This allows us to factor out \(x^5\) from the entire polynomial:

\(x^8 - x^5 = x^5 (x^3 - 1)\)

Clearly identifying the GCF simplifies polynomial factoring. You're breaking the polynomial down into more manageable chunks.

Difference of Cubes

The 'Difference of Cubes' is a special factoring formula used when you encounter a polynomial that fits the form \(a^3 - b^3\).

The difference of cubes can be factored using the formula:

\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Naturally, spotting the 'difference of cubes' pattern speeds up the process of polynomial factoring.

In our example \(x^8 - x^5\), after taking out the GCF, we have \(x^5 (x^3 - 1)\). Now, focus on \(x^3 - 1\) which matches the form \(a^3 - b^3\) with \(a = x\) and \(b = 1\).

Using the difference of cubes formula, we get:

\(x^3 - 1 = (x - 1)(x^2 + x + 1)\).

This step simplifies the polynomial further, making the entire process less intimidating.

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra. It involves expressing a polynomial as a product of its factors.

Let's bring together the steps used for the exercise above.

We began with the polynomial \(x^8 - x^5\). Using the GCF \(x^5\), we factored it to get \(x^5 (x^3 - 1)\).

Next, we recognized that \(x^3 - 1\) could be factored as a difference of cubes:

\(x^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Substituting \(a = x\) and \(b = 1\), we then have:

\(x^3 - 1 = (x - 1)(x^2 + x + 1)\).

Finally, combining all factors, the completely factored form of the polynomial is:

\(x^8 - x^5 = x^5 (x - 1)(x^2 + x + 1)\).

Understanding these steps and recognizing patterns such as the GCF and the difference of cubes will make polynomial factoring much easier. Practice these concepts regularly to build your confidence.