Problem 40 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ a^{4}-b^{4} $$

Step-by-Step Solution

Answer

Answer: The factored form of the binomial \(a^4 - b^4\) is \((a^2 + b^2)(a - b)(a + b)\).

Step 1: Identify the square terms

First, we need to identify the square terms in the binomial. The given binomial is \(a^4 - b^4\), where \(a^4\) and \(b^4\) are the square terms.

Step 2: Apply the difference of squares formula

Now, we can apply the difference of squares formula to factor the binomial: \((a^2 + b^2)(a^2 - b^2)\)

Step 3: Notice that the expressions are still differences of two squares

We can notice that the expressions \((a^2 + b^2)\) and \((a^2 - b^2)\) are still differences of two squares. We can apply the difference of squares formula again to factor these expressions further.

Step 4: Factor \((a^2 + b^2)\) and \((a^2 - b^2)\) using the difference of squares formula

Applying the difference of squares formula to \((a^2 + b^2)\) and \((a^2 - b^2)\), we get: \((a^2 - b^2) = (a - b)(a + b)\) and \((a^2 + b^2)\) remains the same as it is not factored further.

Step 5: Write the final factored form

Now, we can combine the factored expressions from Step 4 to get the final factored form of the given binomial: $$a^4 - b^4 = (a^2 + b^2)(a - b)(a + b)$$ Thus, the factored form of the given binomial is \((a^2 + b^2)(a - b)(a + b)\).