Question 26 of 43intermediate🔍 AnalyzeShort Answer1 mark

Do you think x - y is also a factor of x4x^{4} - y4y^{4}?

Correct Answer

Yes, x - y is a factor of x⁴ - y⁴. We can express x⁴ - y⁴ as (x²)² - (y²)², which equals (x² - y²)(x² + y²). Since we know that x² - y² = (x - y)(x + y), the expression becomes (x - y)(x + y)(x² + y²). Because (x - y) is clearly one of the multiplied terms in the final factorization, it is indeed a factor.

Exercise: Think and Reflect | Q: 1 | (Page 18)
For More Understanding

Explanation

The textbook context demonstrates that x4x^{4} - y4y^{4} can be factored by treating it as a difference of two squares: (x2x^{2})^2 - (y2y^{2})^2. This yields (x2x^{2} - y2y^{2})(x2x^{2} + y2y^{2}). The context also reminds us of the identity x2x^{2} - y2y^{2} = (x - y)(x + y). Substituting this back into the factorization shows that x - y is a factor.

Solution Steps

  1. Step 1: Rewrite x4x^{4} - y4y^{4} as (x2x^{2})^2 - (y2y^{2})^2.

  2. Step 2: Factor the difference of squares to get (x2x^{2} - y2y^{2})(x2x^{2} + y2y^{2}).

  3. Step 3: Substitute (x - y)(x + y) for (x2x^{2} - y2y^{2}).

  4. Step 4: The final expression is (x - y)(x + y)(x2x^{2} + y2y^{2}), proving x - y is a factor.

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