Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(24x2 + 72x) + 81
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 16x2+72x+81
The first term is, 16x2 its coefficient is 16 .
The middle term is, +72x its coefficient is 72 .
The last term, "the constant", is +81
Step-1 : Multiply the coefficient of the first term by the constant 16 • 81 = 1296
Step-2 : Find two factors of 1296 whose sum equals the coefficient of the middle term, which is 72 .
| -1296 | + | -1 | = | -1297 | ||
| -648 | + | -2 | = | -650 | ||
| -432 | + | -3 | = | -435 | ||
| -324 | + | -4 | = | -328 | ||
| -216 | + | -6 | = | -222 | ||
| -162 | + | -8 | = | -170 | ||
| -144 | + | -9 | = | -153 | ||
| -108 | + | -12 | = | -120 | ||
| -81 | + | -16 | = | -97 | ||
| -72 | + | -18 | = | -90 | ||
| -54 | + | -24 | = | -78 | ||
| -48 | + | -27 | = | -75 | ||
| -36 | + | -36 | = | -72 | ||
| -27 | + | -48 | = | -75 | ||
| -24 | + | -54 | = | -78 | ||
| -18 | + | -72 | = | -90 | ||
| -16 | + | -81 | = | -97 | ||
| -12 | + | -108 | = | -120 | ||
| -9 | + | -144 | = | -153 | ||
| -8 | + | -162 | = | -170 | ||
| -6 | + | -216 | = | -222 | ||
| -4 | + | -324 | = | -328 | ||
| -3 | + | -432 | = | -435 | ||
| -2 | + | -648 | = | -650 | ||
| -1 | + | -1296 | = | -1297 | ||
| 1 | + | 1296 | = | 1297 | ||
| 2 | + | 648 | = | 650 | ||
| 3 | + | 432 | = | 435 | ||
| 4 | + | 324 | = | 328 | ||
| 6 | + | 216 | = | 222 | ||
| 8 | + | 162 | = | 170 | ||
| 9 | + | 144 | = | 153 | ||
| 12 | + | 108 | = | 120 | ||
| 16 | + | 81 | = | 97 | ||
| 18 | + | 72 | = | 90 | ||
| 24 | + | 54 | = | 78 | ||
| 27 | + | 48 | = | 75 | ||
| 36 | + | 36 | = | 72 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 36 and 36
16x2 + 36x + 36x + 81
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (4x+9)
Add up the last 2 terms, pulling out common factors :
9 • (4x+9)
Step-5 : Add up the four terms of step 4 :
(4x+9) • (4x+9)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (4x+9) by (4x+9)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (4x+9) and the exponents are :
1 , as (4x+9) is the same number as (4x+9)1
and 1 , as (4x+9) is the same number as (4x+9)1
The product is therefore, (4x+9)(1+1) = (4x+9)2
Final result :
(4x + 9)2
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