Problem 90 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ x^{17}-x^{7} y^{2} $$

Step-by-Step Solution

Answer

\(x^7 (x^5 - y)(x^5 + y)\)

Step 1: Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of the terms in the polynomial. The terms are \(x^{17}\) and \(-x^{7} y^{2}\). Both terms have \(x^7\) as a common factor.

Step 2: Factor out the GCF

Factor \(x^7\) out of both terms. This results in: \(x^7(x^{10} - y^2)\).

Step 3: Recognize the Difference of Squares

Notice that \(x^{10} - y^2\) is a difference of squares. This can be expressed as \((a^2 - b^2)\), where \(a = x^5\) and \(b = y\).

Step 4: Apply the Difference of Squares Formula

Use the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\) to factor \(x^{10} - y^2\): \(x^7 (x^{5} - y)(x^{5} + y)\).

Step 5: Verify the Factorization

Confirm that the factorization is correct by multiplying back: \( x^7(x^5 - y)(x^5 + y) = x^7(x^{10} - y^2) = x^{17} - x^7 y^2\).

Step 6: Identify Prime Polynomials

Check each factor to see if they can be factored further. Both \(x^5 - y\) and \(x^5 + y\) are prime polynomials.

Key Concepts

greatest common factor

The greatest common factor (GCF) is the highest degree of similarity between the terms of a polynomial. To find the GCF in our problem, we look at the terms: \(x^{17}\) and \(-x^{7}y^{2}\). The common factor here is \(x^7\). We choose the smallest power of \(x\) that appears in both terms, which is \(x^7\).

Then, we factor out \(x^7\) from both terms: \(x^{17}\rightarrow x^{17} = x^7(x^{10})\) and \(-x^7 y^2 \rightarrow - x^7 y^2 = x^7(-y^2)\).

This simplifies our polynomial to \(x^7(x^{10} - y^2)\). By finding the GCF, we break down complex polynomials into simpler parts for easier manipulation and solving.

difference of squares

The difference of squares is a special polynomial form \(a^2 - b^2\), which can be factored into \( (a - b)(a + b) \). It's important because it simplifies expressions that follow this pattern.

In our problem, after factoring out the GCF, we have \(x^{10} - y^2\). Notice that \(x^{10}\) is a square of \(x^5\) (\(x^{10} = (x^5)^2\)), and \(y^2\) is obviously \( (y)^2\). So we rewrite it as \( (x^5)^2 - (y)^2 \).

Applying the difference of squares formula:
\[ x^{10} - y^2 = (x^5 - y)(x^5 + y) \]

Therefore, the expression \(x^{7}(x^{10} - y^2)\) can further be factored into \( x^7(x^5 - y)(x^5 + y) \). Recognizing this pattern is crucial in breaking down complex polynomials.

prime polynomials

Prime polynomials are polynomials that cannot be factored any further. They are the 'building blocks' of polynomial expressions.

Once we have our fully factored polynomial \( x^7(x^5 - y)(x^5 + y) \), we need to check if the factors can be broken down more.

In this problem, \((x^5 - y)\) and \((x^5 + y)\) are both prime because there are no common factors and no more expressions of difference of squares or other simple factorizations apply to them.

Identifying prime polynomials ensures that the given polynomial is factored completely, and no further simplification is possible.