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Question Video: Factoring a Higher Degree Expression by Completing the Square Mathematics

Factor 𝑥⁸ − 16𝑦⁸ by completing the square.

04:45

Video Transcript

Factor 𝑥 to the eighth power minus 16𝑦 to the eighth power by completing the square.

In this question, it may not immediately be obvious how we can solve the problem by completing the square. However, we do notice that our expression is of the form 𝑎 squared minus 𝑏 squared. This is known as the difference of squares and can be factored as 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏. Letting 𝑎 squared equal 𝑥 to the eighth power, we know that 𝑎 is equal to 𝑥 to the fourth power. And letting 𝑏 squared equals 16𝑦 to the eighth power, we know that 𝑏 is the square root of this, which is equal to four 𝑦 to the fourth power. The original expression can therefore be rewritten as 𝑥 to the fourth power all squared minus four 𝑦 to the fourth power all squared. And factoring this using the difference of squares, we have 𝑥 to the fourth power plus four 𝑦 to the fourth power multiplied by 𝑥 to the fourth power minus four 𝑦 to the fourth power.

The second part of our expression is once again written in the form 𝑎 squared minus 𝑏 squared. 𝑥 to the fourth power minus four 𝑦 to the fourth power is equal to 𝑥 squared plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑦 squared. These two parentheses cannot be factored any further. So we now need to consider the expression 𝑥 to the fourth power plus four 𝑦 to the fourth power. It is this expression that we’ll be able to factor by completing the square. This is in the form 𝑐 squared plus 𝑑 squared. And we need to manipulate this so it is in the form of a perfect square trinomial.

We know that any perfect square trinomial of the form 𝑐 squared plus two 𝑐𝑑 plus 𝑑 squared can be factored into the form 𝑐 plus 𝑑 all squared. Since 𝑐 squared is equal to 𝑥 to the fourth power, 𝑐 is equal to 𝑥 squared. Likewise, since 𝑑 squared is equal to four 𝑦 to the fourth power, 𝑑 is equal to two 𝑦 squared. The term two 𝑐𝑑 is therefore equal to two multiplied by 𝑥 squared multiplied by two 𝑦 squared. And this is equal to four 𝑥 squared 𝑦 squared.

We need to add the positive and negative of this to the expression in our first set of parentheses to create our perfect square trinomial. The full expression can be written as shown. And we can now factor the perfect square trinomial contained in the first set of parentheses. This is equal to 𝑥 squared plus two 𝑦 squared all squared. And rewriting four 𝑥 squared 𝑦 squared as two 𝑥𝑦 all squared, we have 𝑥 squared plus two 𝑦 squared all squared minus two 𝑥𝑦 all squared multiplied by 𝑥 squared plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑦 squared.

We note that the expression within the brackets is the difference of squares. And using the rule we saw earlier, this factors to 𝑥 squared plus two 𝑦 squared plus two 𝑥𝑦 multiplied by 𝑥 squared plus two 𝑦 squared minus two 𝑥𝑦. We can then reorder the terms in these parentheses as shown, giving us a final answer of 𝑥 squared plus two 𝑥𝑦 plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑥𝑦 plus two 𝑦 squared multiplied by 𝑥 squared plus two 𝑦 squared multiplied by 𝑥 squared minus two 𝑦 squared. This is the fully factored form of 𝑥 to the eighth power minus 16𝑦 to the eighth power.

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