SOLUTION: How do you factor the following problems step by step? 1. 9-x^2 2. 4x^2+81-36x 3. 4y^2+16y+16 4. x^-25/x+5

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: How do you factor the following problems step by step? 1. 9-x^2 2. 4x^2+81-36x 3. 4y^2+16y+16 4. x^-25/x+5      Log On


   

Question 133168: How do you factor the following problems step by step?
1. 9-x^2
2. 4x^2+81-36x
3. 4y^2+16y+16
4. x^-25/x+5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started


# 1

9-x%5E2 Start with the given expression

%283%29%5E2-x%5E2 Rewrite 9 as %283%29%5E2

%283%29%5E2-%28x%29%5E2 Rewrite x%5E2 as %28x%29%5E2


Now use the difference of squares. Remember, the difference of squares formula is A%5E2-B%5E2=%28A%2BB%29%28A-B%29 where in this case A=3 and B=x

9-x%5E2=%283%2Bx%29%283-x%29 Plug in A=3 and B=x



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Answer:

So the expression

9-x%5E2

factors to

%283%2Bx%29%283-x%29

Notice that if you foil the factored expression, you get the original expression. This verifies our answer.




4x%5E2%2B81-36x Start with the given expression


4x%5E2-36x%2B81 Rearrange the terms




Looking at 4x%5E2-36x%2B81 we can see that the first term is 4x%5E2 and the last term is 81 where the coefficients are 4 and 81 respectively.

Now multiply the first coefficient 4 and the last coefficient 81 to get 324. Now what two numbers multiply to 324 and add to the middle coefficient -36? Let's list all of the factors of 324:



Factors of 324:
1,2,3,4,6,9,12,18,27,36,54,81,108,162

-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 324
1*324
2*162
3*108
4*81
6*54
9*36
12*27
18*18
(-1)*(-324)
(-2)*(-162)
(-3)*(-108)
(-4)*(-81)
(-6)*(-54)
(-9)*(-36)
(-12)*(-27)
(-18)*(-18)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36

First NumberSecond NumberSum
13241+324=325
21622+162=164
31083+108=111
4814+81=85
6546+54=60
9369+36=45
122712+27=39
181818+18=36
-1-324-1+(-324)=-325
-2-162-2+(-162)=-164
-3-108-3+(-108)=-111
-4-81-4+(-81)=-85
-6-54-6+(-54)=-60
-9-36-9+(-36)=-45
-12-27-12+(-27)=-39
-18-18-18+(-18)=-36



From this list we can see that -18 and -18 add up to -36 and multiply to 324


Now looking at the expression 4x%5E2-36x%2B81, replace -36x with -18x%2B-18x (notice -18x%2B-18x adds up to -36x. So it is equivalent to -36x)

4x%5E2%2Bhighlight%28-18x%2B-18x%29%2B81


Now let's factor 4x%5E2-18x-18x%2B81 by grouping:


%284x%5E2-18x%29%2B%28-18x%2B81%29 Group like terms


2x%282x-9%29-9%282x-9%29 Factor out the GCF of 2x out of the first group. Factor out the GCF of -9 out of the second group


%282x-9%29%282x-9%29 Since we have a common term of 2x-9, we can combine like terms

So 4x%5E2-18x-18x%2B81 factors to %282x-9%29%282x-9%29


So this also means that 4x%5E2-36x%2B81 factors to %282x-9%29%282x-9%29 (since 4x%5E2-36x%2B81 is equivalent to 4x%5E2-18x-18x%2B81)


note: %282x-9%29%282x-9%29 is equivalent to %282x-9%29%5E2 since the term 2x-9 occurs twice. So 4x%5E2-36x%2B81 also factors to %282x-9%29%5E2



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Answer:
So 4x%5E2-36x%2B81 factors to %282x-9%29%5E2