Problem 70 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$7 x^{4}-7$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(7x^{4} - 7\) is \(7(x^{2} + 1)(x+1)(x-1)\)

Step 1: Identify and Factor out the Greatest Common Factor (GCF)

In the given polynomial \(7x^{4} - 7\), the number 7 is a common factor. Factoring out 7 gives \(7(x^{4} - 1)\)

Step 2: Identify the Difference of Squares

The expression \(x^{4} - 1\) can be seen as a difference of squares. Remember that a difference of squares \(a^{2}-b^{2}\) can be factored as \((a+b)(a-b)\). Here \(a = x^{2}\) and \(b = 1\). Therefore, \(x^{4} - 1\) can be factored as \((x^{2} + 1)(x^{2} - 1)\).

Step 3: Continue Factoring if Possible

In this case, \(x^{2} - 1\) is a difference of squares, which could be further factored. By applying the difference of squares formula again, it can be factored as \((x+1)(x-1)\). However, \(x^{2} + 1\) can't be factored using real numbers. Any attempt to do so would yield complex roots. Thus, it remains as is.

Step 4: Write the Fully Factored Form

Multiplying the common factor we found in Step 1 with the factored form, we find the expression factors to \(7(x^{2} + 1)(x+1)(x-1)\).