SOLUTION: Factor the following polynomial completely. (If the polynomial cannot be factored, enter PRIME.) -x^2+16x+57 Someone help me solve this.

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Question 319179: Factor the following polynomial completely. (If the polynomial cannot be factored, enter PRIME.)
-x^2+16x+57
Someone help me solve this.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-x%5E2%2B16x%2B57 Start with the given expression.


-%28x%5E2-16x-57%29 Factor out the GCF -1.


Now let's try to factor the inner expression x%5E2-16x-57


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Looking at the expression x%5E2-16x-57, we can see that the first coefficient is 1, the second coefficient is -16, and the last term is -57.


Now multiply the first coefficient 1 by the last term -57 to get %281%29%28-57%29=-57.


Now the question is: what two whole numbers multiply to -57 (the previous product) and add to the second coefficient -16?


To find these two numbers, we need to list all of the factors of -57 (the previous product).


Factors of -57:
1,3,19,57
-1,-3,-19,-57


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -57.
1*(-57) = -57
3*(-19) = -57
(-1)*(57) = -57
(-3)*(19) = -57

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -16:


First NumberSecond NumberSum
1-571+(-57)=-56
3-193+(-19)=-16
-157-1+57=56
-319-3+19=16



From the table, we can see that the two numbers 3 and -19 add to -16 (the middle coefficient).


So the two numbers 3 and -19 both multiply to -57 and add to -16


Now replace the middle term -16x with 3x-19x. Remember, 3 and -19 add to -16. So this shows us that 3x-19x=-16x.


x%5E2%2Bhighlight%283x-19x%29-57 Replace the second term -16x with 3x-19x.


%28x%5E2%2B3x%29%2B%28-19x-57%29 Group the terms into two pairs.


x%28x%2B3%29%2B%28-19x-57%29 Factor out the GCF x from the first group.


x%28x%2B3%29-19%28x%2B3%29 Factor out 19 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28x-19%29%28x%2B3%29 Combine like terms. Or factor out the common term x%2B3


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So -1%28x%5E2-16x-57%29 then factors further to -%28x-19%29%28x%2B3%29


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Answer:


So -x%5E2%2B16x%2B57 completely factors to -%28x-19%29%28x%2B3%29.


In other words, -x%5E2%2B16x%2B57=-%28x-19%29%28x%2B3%29.


Note: you can check the answer by expanding -%28x-19%29%28x%2B3%29 to get -x%5E2%2B16x%2B57 or by graphing the original expression and the answer (the two graphs should be identical).