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Start with the given expression.
Rearrange the terms in descending exponential order.
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
:
1,3,5,9,15,25,27,45,75,81,135,225,405,675,2025
-1,-3,-5,-9,-15,-25,-27,-45,-75,-81,-135,-225,-405,-675,-2025
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to
.
1*2025
3*675
5*405
9*225
15*135
25*81
27*75
45*45
(-1)*(-2025)
(-3)*(-675)
(-5)*(-405)
(-9)*(-225)
(-15)*(-135)
(-25)*(-81)
(-27)*(-75)
(-45)*(-45)
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 | 2025 | 1+2025=2026 |
| 3 | 675 | 3+675=678 |
| 5 | 405 | 5+405=410 |
| 9 | 225 | 9+225=234 |
| 15 | 135 | 15+135=150 |
| 25 | 81 | 25+81=106 |
| 27 | 75 | 27+75=102 |
| 45 | 45 | 45+45=90 |
| -1 | -2025 | -1+(-2025)=-2026 |
| -3 | -675 | -3+(-675)=-678 |
| -5 | -405 | -5+(-405)=-410 |
| -9 | -225 | -9+(-225)=-234 |
| -15 | -135 | -15+(-135)=-150 |
| -25 | -81 | -25+(-81)=-106 |
| -27 | -75 | -27+(-75)=-102 |
| -45 | -45 | -45+(-45)=-90 |
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
---------------------------------------------
Answer:
So
factors to
.
In other words,
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).
(