You can
put this solution on YOUR website!# 6
Looking at the expression
, we can see that the first coefficient is
, the second coefficient is
, and the last coefficient is
.
Now multiply the first coefficient
by the last coefficient
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
:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to
.
1*1764 = 1764
2*882 = 1764
3*588 = 1764
4*441 = 1764
6*294 = 1764
7*252 = 1764
9*196 = 1764
12*147 = 1764
14*126 = 1764
18*98 = 1764
21*84 = 1764
28*63 = 1764
36*49 = 1764
42*42 = 1764
(-1)*(-1764) = 1764
(-2)*(-882) = 1764
(-3)*(-588) = 1764
(-4)*(-441) = 1764
(-6)*(-294) = 1764
(-7)*(-252) = 1764
(-9)*(-196) = 1764
(-12)*(-147) = 1764
(-14)*(-126) = 1764
(-18)*(-98) = 1764
(-21)*(-84) = 1764
(-28)*(-63) = 1764
(-36)*(-49) = 1764
(-42)*(-42) = 1764
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 | 1764 | 1+1764=1765 |
| 2 | 882 | 2+882=884 |
| 3 | 588 | 3+588=591 |
| 4 | 441 | 4+441=445 |
| 6 | 294 | 6+294=300 |
| 7 | 252 | 7+252=259 |
| 9 | 196 | 9+196=205 |
| 12 | 147 | 12+147=159 |
| 14 | 126 | 14+126=140 |
| 18 | 98 | 18+98=116 |
| 21 | 84 | 21+84=105 |
| 28 | 63 | 28+63=91 |
| 36 | 49 | 36+49=85 |
| 42 | 42 | 42+42=84 |
| -1 | -1764 | -1+(-1764)=-1765 |
| -2 | -882 | -2+(-882)=-884 |
| -3 | -588 | -3+(-588)=-591 |
| -4 | -441 | -4+(-441)=-445 |
| -6 | -294 | -6+(-294)=-300 |
| -7 | -252 | -7+(-252)=-259 |
| -9 | -196 | -9+(-196)=-205 |
| -12 | -147 | -12+(-147)=-159 |
| -14 | -126 | -14+(-126)=-140 |
| -18 | -98 | -18+(-98)=-116 |
| -21 | -84 | -21+(-84)=-105 |
| -28 | -63 | -28+(-63)=-91 |
| -36 | -49 | -36+(-49)=-85 |
| -42 | -42 | -42+(-42)=-84 |
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.
Factor out the GCF
Condense the terms.
So
completely factors to
In other words,
=======================================================
# 7
Start with the given expression.
Rewrite
as
.
Rewrite
as
.
Notice how we have a difference of squares
where in this case
and
.
So let's use the difference of squares formula
to factor the expression:
Start with the difference of squares formula.
Plug in
and
.
So this shows us that
factors to
.
In other words
.
Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at
jim_thompson5910@hotmail.com
or you can visit my website here:
http://www.freewebs.com/jimthompson5910/home.html
Thanks,
Jim
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