SOLUTION: Factor the polynomial x2 + 10x + 9. Enter each factor as a polynomial in descending order.

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Question 435729: Factor the polynomial x2 + 10x + 9. Enter each factor as a polynomial in descending order.
Found 3 solutions by Alan3354, rwm, richard1234:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Hint: 9 + 1 = 10

Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B10x%2B9, we can see that the first coefficient is 1, the second coefficient is 10, and the last term is 9.



Now multiply the first coefficient 1 by the last term 9 to get %281%29%289%29=9.



Now the question is: what two whole numbers multiply to 9 (the previous product) and add to the second coefficient 10?



To find these two numbers, we need to list all of the factors of 9 (the previous product).



Factors of 9:

1,3,9

-1,-3,-9



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



These factors pair up and multiply to 9.

1*9 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:



First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6




From the table, we can see that the two numbers 1 and 9 add to 10 (the middle coefficient).



So the two numbers 1 and 9 both multiply to 9 and add to 10



Now replace the middle term 10x with x%2B9x. Remember, 1 and 9 add to 10. So this shows us that x%2B9x=10x.



x%5E2%2Bhighlight%28x%2B9x%29%2B9 Replace the second term 10x with x%2B9x.



%28x%5E2%2Bx%29%2B%289x%2B9%29 Group the terms into two pairs.



x%28x%2B1%29%2B%289x%2B9%29 Factor out the GCF x from the first group.



x%28x%2B1%29%2B9%28x%2B1%29 Factor out 9 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%2B9%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1



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Answer:



So x%5E2%2B10%2Ax%2B9 factors to %28x%2B9%29%28x%2B1%29.



In other words, x%5E2%2B10%2Ax%2B9=%28x%2B9%29%28x%2B1%29.



Note: you can check the answer by expanding %28x%2B9%29%28x%2B1%29 to get x%5E2%2B10%2Ax%2B9 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
We want to factor it in the form (x+a)(x+b). Since ab = 9 and a+b = 10, we can easily obtain {a,b} = {1,9} and the polynomial factors to (x+9)(x+1).