Question 28 of 43intermediate🔍 AnalyzeShort Answer1 mark

How about x5x^{5} - y5y^{5}? Does this also have x - y as a factor?

Correct Answer

Yes, x5−y5x^5 - y^5 also has x−yx - y as a factor. Following the pattern where x−yx - y is a common factor of x2−y2x^2 - y^2, x3−y3x^3 - y^3, and x4−y4x^4 - y^4, it is also a factor of x5−y5x^5 - y^5.

Exercise: Think and Reflect | Q: 3 | (Page 18)
For More Understanding

Explanation

The textbook context establishes a pattern by showing that x−yx - y is a factor of x2−y2x^2 - y^2 and x3−y3x^3 - y^3. It then demonstrates that x−yx - y is also a factor of x4−y4x^4 - y^4 by expressing it as (x2−y2)(x2+y2)(x^2 - y^2)(x^2 + y^2). By extending this observed pattern, students are prompted to realize that x−yx - y will similarly be a factor of x5−y5x^5 - y^5.

Solution Steps

  1. Step 1: Observe the pattern from the given identities: x−yx - y is a common factor of x2−y2x^2 - y^2 and x3−y3x^3 - y^3.

  2. Step 2: Note that x−yx - y is also a factor of x4−y4x^4 - y^4, as shown by the relation x4−y4=(x2−y2)(x2+y2)x^4 - y^4 = (x^2 - y^2)(x^2 + y^2).

  3. Step 3: Conclude based on this established pattern that x−yx - y is also a factor of x5−y5x^5 - y^5.

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