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Chapter 0: Problem 70

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

Short Answer

Expert verified
The completely factored form of the expression is \(7(x - 1)(x + 1)(x^{2} + 1)\)

Step by step solution

01

Factor Out the Greatest Common Factor

The first step in factoring is always to factor out the greatest common factor (GCF). The GCF of \(7x^{4}\) and \(7\) is \(7\). So, we can factor out 7 from the original polynomial. This gives us: \(7(x^{4} - 1)\)
02

Factor the Difference of Squares

The expression \(x^{4} - 1\) is a difference of squares, which can be factored as \((x^{2} - 1)(x^{2} + 1)\). Thus, our expression now becomes: \(7(x^{2} - 1)(x^{2} + 1)\)
03

Factor the Difference of Squares Again

The expression \(x^{2} - 1\) is also a difference of squares, which can be factored as \((x - 1)(x + 1)\). Thus, after this step, our expression becomes: \(7(x - 1)(x + 1)(x^{2} + 1)\)
04

Factor the Sum of Squares

The expression \(x^{2} + 1\) is a sum of squares. It cannot be factored into simple factors using real numbers, so it remains as is.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

What is a compound inequality and how is it solved?

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?
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