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Chapter 0: Problem 134

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{3}}=-2$$

Short Answer

Expert verified
The statement is false. The correct factorization for the polynomial \(x^{4}-16\) is \(\left(x^{2}+4\right)\left(x-2\right)\left(x+2\right)\)

Step by step solution

01

Evaluation

Check the expression \(\left(x^{2}+4\right)\left(x^{2}-4\right)\), when multiplied it gives \(x^{4}-16x^{2}+16\), which is not equal to \(x^{4}-16\). Hence, the statement is false.
02

Correct Factorization

Factorize the polynomial \(x^{4}-16\), using the formula for the difference of squares which states that \(a^{2}-b^{2}=(a+b)(a-b)\). When applied to the given exercise, it factors the equation into \(\left(x^{2}+4\right)\left(x^{2}-4\right)\). However, \(x^{2}-4\) can be further factored into \(x+2\) and \(x-2\).
03

Final Solution

So, the correct factorization for \(x^{4}-16\) is \(\left(x^{2}+4\right)\left(x-2\right)\left(x+2\right)\).

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