Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{7}-x^{5} $$

Short answer

\[x^7 - x^5 = x^5(x - 1)(x + 1)\]

Step-by-step solution

Identify the Greatest Common Factor (GCF)

Look for the greatest common factor in the terms of the polynomial. In this case, both terms contain a factor of \(x^5\).

Factor out the GCF

Factor \(x^5\) out of each term: \[x^7 - x^5 = x^5(x^2 - 1)\]

Recognize the difference of squares

Notice that \(x^2 - 1\) is a difference of squares, which can be factored further. The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).

Factor the difference of squares

Applying the difference of squares formula to \(x^2 - 1\) we get: \[x^2 - 1 = (x - 1)(x + 1)\]

Combine the factors

Putting it all together, we get: \[x^7 - x^5 = x^5(x - 1)(x + 1)\]

Key concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides all terms in a polynomial. Think of it as the biggest 'chunk' shared by every term. For example, in the polynomial given, \(x^7 - x^5\), both terms have a common factor of \(x^5\). Finding the GCF helps in simplifying the problem. By factoring out the GCF, we get:
ote: After factoring \(x^5\) out of \(x^7 - x^5\), we are left with:
\text{\(x^7 - x^5 = x^5(x^2 - 1)\)}.
This step reduces the polynomial into a simpler form, making it easier to factor further. Remember, always start by finding the GCF when factoring polynomials.

Difference of Squares

The difference of squares is a special algebraic form where we subtract two square terms. The general formula for factoring a difference of squares is: \((a^2 - b^2) = (a - b)(a + b)\). In our exercise, the term inside the parentheses, after factoring out the \(x^5\), is \(x^2 - 1\). Notice that this matches the difference of squares pattern where \(a = x\) and \(b = 1\).
Applying the difference of squares formula, we get:

ote: \(x^2 - 1\) becomes \((x - 1)(x + 1)\).
Using this technique allows you to further break down the polynomial into simpler binomials.

Algebraic Factoring Techniques

Algebraic factoring techniques involve breaking down polynomials into simpler factors. These techniques include:

• Factoring out the GCF
• Recognizing special patterns: like the difference of squares, perfect square trinomials, and sum/difference of cubes.
• Using trial and error: For trinomials, looking for pairs that multiply to give the product of the extreme coefficients and add to the middle coefficient.
• Grouping: For polynomials with four or more terms, grouping can help find common factors.

In our example, we applied two key techniques: factoring out the GCF and using the difference of squares. Here is the full process again:

ote: 1. Factored out the GCF: \(x^5\)
ote: 2. Recognized the difference of squares in \(x^2 - 1\) and factored it as \((x - 1)(x + 1)\).
ote: The final factored form is:
ote: \(x^5(x - 1)(x + 1)\)
Understanding these basic techniques will help you tackle various polynomial factoring problems effectively.