SOLUTION: Please help me factor this expression 36x^4-8x^3-28x^2

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please help me factor this expression 36x^4-8x^3-28x^2      Log On


   

Question 315391: Please help me factor this expression 36x^4-8x^3-28x^2
Found 2 solutions by jim_thompson5910, mananth:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

36x%5E4-8x%5E3-28x%5E2 Start with the given expression.


4x%5E2%289x%5E2-2x-7%29 Factor out the GCF 4x%5E2.


Now let's try to factor the inner expression 9x%5E2-2x-7


---------------------------------------------------------------


Looking at the expression 9x%5E2-2x-7, we can see that the first coefficient is 9, the second coefficient is -2, and the last term is -7.


Now multiply the first coefficient 9 by the last term -7 to get %289%29%28-7%29=-63.


Now the question is: what two whole numbers multiply to -63 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -63 (the previous product).


Factors of -63:
1,3,7,9,21,63
-1,-3,-7,-9,-21,-63


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


These factors pair up and multiply to -63.
1*(-63) = -63
3*(-21) = -63
7*(-9) = -63
(-1)*(63) = -63
(-3)*(21) = -63
(-7)*(9) = -63

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
1-631+(-63)=-62
3-213+(-21)=-18
7-97+(-9)=-2
-163-1+63=62
-321-3+21=18
-79-7+9=2



From the table, we can see that the two numbers 7 and -9 add to -2 (the middle coefficient).


So the two numbers 7 and -9 both multiply to -63 and add to -2


Now replace the middle term -2x with 7x-9x. Remember, 7 and -9 add to -2. So this shows us that 7x-9x=-2x.


9x%5E2%2Bhighlight%287x-9x%29-7 Replace the second term -2x with 7x-9x.


%289x%5E2%2B7x%29%2B%28-9x-7%29 Group the terms into two pairs.


x%289x%2B7%29%2B%28-9x-7%29 Factor out the GCF x from the first group.


x%289x%2B7%29-1%289x%2B7%29 Factor out 1 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-1%29%289x%2B7%29 Combine like terms. Or factor out the common term 9x%2B7


--------------------------------------------------


So 4x%5E2%289x%5E2-2x-7%29 then factors further to 4x%5E2%28x-1%29%289x%2B7%29


===============================================================


Answer:


So 36x%5E4-8x%5E3-28x%5E2 completely factors to 4x%5E2%28x-1%29%289x%2B7%29.


In other words, 36x%5E4-8x%5E3-28x%5E2=4x%5E2%28x-1%29%289x%2B7%29.


Note: you can check the answer by expanding 4x%5E2%28x-1%29%289x%2B7%29 to get 36x%5E4-8x%5E3-28x%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
36x^4-8x^3-28x^2
4x^2(9x^2-2x-7)
4x^2(9x^2-9x+7x-7)
4x^2{9x(x-1)+7(x-1)}
4x^2(9x+7)(x-1)