Problem 134 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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 1 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.

Step 2 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 \(x-2\).

Step 3 Final Solution

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