Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
((22•32n2) + 60n) + 25
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 36n2+60n+25
The first term is, 36n2 its coefficient is 36 .
The middle term is, +60n its coefficient is 60 .
The last term, "the constant", is +25
Step-1 : Multiply the coefficient of the first term by the constant 36 • 25 = 900
Step-2 : Find two factors of 900 whose sum equals the coefficient of the middle term, which is 60 .
| -900 | + | -1 | = | -901 | ||
| -450 | + | -2 | = | -452 | ||
| -300 | + | -3 | = | -303 | ||
| -225 | + | -4 | = | -229 | ||
| -180 | + | -5 | = | -185 | ||
| -150 | + | -6 | = | -156 | ||
| -100 | + | -9 | = | -109 | ||
| -90 | + | -10 | = | -100 | ||
| -75 | + | -12 | = | -87 | ||
| -60 | + | -15 | = | -75 | ||
| -50 | + | -18 | = | -68 | ||
| -45 | + | -20 | = | -65 | ||
| -36 | + | -25 | = | -61 | ||
| -30 | + | -30 | = | -60 | ||
| -25 | + | -36 | = | -61 | ||
| -20 | + | -45 | = | -65 | ||
| -18 | + | -50 | = | -68 | ||
| -15 | + | -60 | = | -75 | ||
| -12 | + | -75 | = | -87 | ||
| -10 | + | -90 | = | -100 | ||
| -9 | + | -100 | = | -109 | ||
| -6 | + | -150 | = | -156 | ||
| -5 | + | -180 | = | -185 | ||
| -4 | + | -225 | = | -229 | ||
| -3 | + | -300 | = | -303 | ||
| -2 | + | -450 | = | -452 | ||
| -1 | + | -900 | = | -901 | ||
| 1 | + | 900 | = | 901 | ||
| 2 | + | 450 | = | 452 | ||
| 3 | + | 300 | = | 303 | ||
| 4 | + | 225 | = | 229 | ||
| 5 | + | 180 | = | 185 | ||
| 6 | + | 150 | = | 156 | ||
| 9 | + | 100 | = | 109 | ||
| 10 | + | 90 | = | 100 | ||
| 12 | + | 75 | = | 87 | ||
| 15 | + | 60 | = | 75 | ||
| 18 | + | 50 | = | 68 | ||
| 20 | + | 45 | = | 65 | ||
| 25 | + | 36 | = | 61 | ||
| 30 | + | 30 | = | 60 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 30 and 30
36n2 + 30n + 30n + 25
Step-4 : Add up the first 2 terms, pulling out like factors :
6n • (6n+5)
Add up the last 2 terms, pulling out common factors :
5 • (6n+5)
Step-5 : Add up the four terms of step 4 :
(6n+5) • (6n+5)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (6n+5) by (6n+5)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (6n+5) and the exponents are :
1 , as (6n+5) is the same number as (6n+5)1
and 1 , as (6n+5) is the same number as (6n+5)1
The product is therefore, (6n+5)(1+1) = (6n+5)2
Final result :
(6n + 5)2
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