SOLUTION: I need to factor completely 209) 16x2 – 40x + 25 210) 2y3 – 16 211) x4 + 125x

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Question 177416: I need to factor completely
209) 16x2 – 40x + 25

210) 2y3 – 16

211) x4 + 125x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
209)



Looking at the expression 16x%5E2-40x%2B25, we can see that the first coefficient is 16, the second coefficient is -40, and the last term is 25.


Now multiply the first coefficient 16 by the last term 25 to get %2816%29%2825%29=400.


Now the question is: what two whole numbers multiply to 400 (the previous product) and add to the second coefficient -40?


To find these two numbers, we need to list all of the factors of 400 (the previous product).


Factors of 400:
1,2,4,5,8,10,16,20,25,40,50,80,100,200,400
-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200,-400


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


These factors pair up and multiply to 400.
1*400
2*200
4*100
5*80
8*50
10*40
16*25
20*20
(-1)*(-400)
(-2)*(-200)
(-4)*(-100)
(-5)*(-80)
(-8)*(-50)
(-10)*(-40)
(-16)*(-25)
(-20)*(-20)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -40:


First NumberSecond NumberSum
14001+400=401
22002+200=202
41004+100=104
5805+80=85
8508+50=58
104010+40=50
162516+25=41
202020+20=40
-1-400-1+(-400)=-401
-2-200-2+(-200)=-202
-4-100-4+(-100)=-104
-5-80-5+(-80)=-85
-8-50-8+(-50)=-58
-10-40-10+(-40)=-50
-16-25-16+(-25)=-41
-20-20-20+(-20)=-40



From the table, we can see that the two numbers -20 and -20 add to -40 (the middle coefficient).


So the two numbers -20 and -20 both multiply to 400 and add to -40


Now replace the middle term -40x with -20x-20x. Remember, -20 and -20 add to -40. So this shows us that -20x-20x=-40x.


16x%5E2%2Bhighlight%28-20x-20x%29%2B25 Replace the second term -40x with -20x-20x.


%2816x%5E2-20x%29%2B%28-20x%2B25%29 Group the terms into two pairs.


4x%284x-5%29%2B%28-20x%2B25%29 Factor out the GCF 4x from the first group.


4x%284x-5%29-5%284x-5%29 Factor out 5 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.


%284x-5%29%284x-5%29 Combine like terms. Or factor out the common term 4x-5


%284x-5%29%5E2 Condense the terms

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Answer:


So 16x%5E2-40x%2B25 factors to %284x-5%29%5E2.


Note: you can check the answer by FOILing %284x-5%29%5E2 to get 16x%5E2-40x%2B25 or by graphing the original expression and the answer (the two graphs should be identical).






210)



2y%5E3-16 Start with the given expression


2%28y%5E3-8%29 Factor out the GCF 2


Now let's focus on the inner expression y%5E3-8


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y%5E3-8 Start with the given expression.


%28y%29%5E3-%282%29%5E3 Rewrite y%5E3 as %28y%29%5E3. Rewrite 8 as %282%29%5E3.


%28y-2%29%28%28y%29%5E2%2B%28y%29%282%29%2B%282%29%5E2%29 Now factor by using the difference of cubes formula. Remember the difference of cubes formula is A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29


%28y-2%29%28y%5E2%2B2y%2B4%29 Multiply

So y%5E3-8 factors to %28y-2%29%28y%5E2%2B2y%2B4%29.

In other words, y%5E3-8=%28y-2%29%28y%5E2%2B2y%2B4%29

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So this means that 2%28y%5E3-8%29 factors down to 2%28y-2%29%28y%5E2%2B2y%2B4%29


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Answer:


So 2y%5E3-16 completely factors to 2%28y-2%29%28y%5E2%2B2y%2B4%29






211)


x%5E4%2B125x Start with the given expression


x%28x%5E3%2B125%29 Factor out the GCF x


Now let's focus on the inner expression x%5E3%2B125

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x%5E3%2B125 Start with the given expression.


%28x%29%5E3%2B%285%29%5E3 Rewrite x%5E3 as %28x%29%5E3. Rewrite 125 as %285%29%5E3.


%28x%2B5%29%28%28x%29%5E2-%28x%29%285%29%2B%285%29%5E2%29 Now factor by using the sum of cubes formula. Remember the sum of cubes formula is A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29


%28x%2B5%29%28x%5E2-5x%2B25%29 Multiply


So x%5E3%2B125 factors to %28x%2B5%29%28x%5E2-5x%2B25%29.

In other words, x%5E3%2B125=%28x%2B5%29%28x%5E2-5x%2B25%29

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So x%28x%5E3%2B125%29 factors to x%28x%2B5%29%28x%5E2-5x%2B25%29

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Answer:

So x%5E4%2B125x completely factors to x%28x%2B5%29%28x%5E2-5x%2B25%29