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

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

210) 2y3 � 16

211) x4 + 125x

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
209)

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

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

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

Factors of :
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 .
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 :

First Number	Second Number	Sum
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
-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 and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out 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.

Combine like terms. Or factor out the common term

Condense the terms

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

Answer:

So factors to .

Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

210)

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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

Start with the given expression.

Rewrite as . Rewrite as .

Now factor by using the difference of cubes formula. Remember the difference of cubes formula is

Multiply

So factors to .

In other words,

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

So this means that factors down to

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

Answer:

So completely factors to

211)

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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

Start with the given expression.

Rewrite as . Rewrite as .

Now factor by using the sum of cubes formula. Remember the sum of cubes formula is

Multiply

So factors to .

In other words,

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

So factors to

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

Answer:

So completely factors to
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