Factor each polynomial. See Examples 5 and 6.

Question

x^{2} - 8 x + 16 - y^{2}

Accepted Answer

Hey, everyone here, we are asked to completely factor the expression, each squared minus 20 H plus minus K squared. Now our first step here in factoring this expression is to just group together these first three terms. So we have the quantity of H squared minus H plus 100. And then outside of the parentheses, we have minus K squared. So now we just want to focus on the grouped terms here and we can see that these terms are a perfect square, try no meal because they follow the perfect square, try no meal formula, which states that X squared minus two, two XY plus Y squared is equal to the quantity of X minus Y squared. So now we can use this formula and find our values for X and Y by comparing it to the grouped terms from our original expression. So doing this, we can start by finding X. So from the formula, we see that X squared is the first term. So looking at our expression, we see that X squared is H squared, which tells us that X is just going to be a church. And now for why when we see that in the formula, why squared is the third term? So the third term in our expression is 100. Therefore, why squared is 100? Which tells us that the value of why is just going to be 10? So now with these values for X and Y identified, we can go ahead and just plug them into our formula here to find our factored expression. So again, our grouped terms, we had H squared minus 20 H plus 100. And now using the formula, we can factor this expression here, which will end up being the quantity of X, which is H minus Y, which is 10 quantity, this whole quantity squared. And so with this factory ization here, with this factored expression, we can go ahead and rewrite our original expression. So if we recall the original expression, we had the quantity of H squared minus 20 H plus 100 then minus K squared. And now again, using our factored form for the group terms, we can rewrite the expression as the quantity of H minus squared minus K squared. So now looking at our rewritten expression, we can see that we actually have the difference of two squares here. So we need to recall the difference of squares formula which tells us that X squared minus Y squared is equal to the quantity of X plus Y times the quantity of X minus Y. So now just like before we need to uh identify the values for X and Y in relation to our expression here. So looking at our formula, we see that the first term is X squared and looking at our expression, we see that X squared, our first term here is the quantity of H minus 10 squared, which tells us that X is just going to be a church minus 10. So now finding our value for why we see that the second term is going to be Y squared. And in our expression, why squared, our second term is K squared. Therefore, we know that the value for Y is just going to be K. And now we can use these values for X and Y and plug them into our formula to find our factored expression here. So again, we have the difference of two squares and using the formula, we have the quantity of H minus plus K times the quantity of H minus 10 minus K. And with this, we have fully factored our original expression which leaves us with answer choice. D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Factor each polynomial. See Examples 5 and 6. $x^{2} - 8 x + 16 - y^{2}$

Key Concepts

Factoring Quadratic Trinomials

Difference of Squares

Combining Factoring Techniques

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