SOLUTION: factor x^4-11x^3-12x^2

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Question 538206: factor x^4-11x^3-12x^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
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x%5E4-11x%5E3-12x%5E2 Start with the given expression.


x%5E2%28x%5E2-11x-12%29 Factor out the GCF x%5E2.


Now let's try to factor the inner expression x%5E2-11x-12


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Looking at the expression x%5E2-11x-12, we can see that the first coefficient is 1, the second coefficient is -11, and the last term is -12.


Now multiply the first coefficient 1 by the last term -12 to get %281%29%28-12%29=-12.


Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of -12 (the previous product).


Factors of -12:
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


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


These factors pair up and multiply to -12.
1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -11:


First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1



From the table, we can see that the two numbers 1 and -12 add to -11 (the middle coefficient).


So the two numbers 1 and -12 both multiply to -12 and add to -11


Now replace the middle term -11x with x-12x. Remember, 1 and -12 add to -11. So this shows us that x-12x=-11x.


x%5E2%2Bhighlight%28x-12x%29-12 Replace the second term -11x with x-12x.


%28x%5E2%2Bx%29%2B%28-12x-12%29 Group the terms into two pairs.


x%28x%2B1%29%2B%28-12x-12%29 Factor out the GCF x from the first group.


x%28x%2B1%29-12%28x%2B1%29 Factor out 12 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-12%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1


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So x%5E2%28x%5E2-11x-12%29 then factors further to x%5E2%28x-12%29%28x%2B1%29


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Answer:


So x%5E4-11x%5E3-12x%5E2 completely factors to x%5E2%28x-12%29%28x%2B1%29.


In other words, x%5E4-11x%5E3-12x%5E2=x%5E2%28x-12%29%28x%2B1%29.


Note: you can check the answer by expanding x%5E2%28x-12%29%28x%2B1%29 to get x%5E4-11x%5E3-12x%5E2 or by graphing the original expression and the answer (the two graphs should be identical).


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Jim