Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(24x2 + 40x) + 25
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 16x2+40x+25
The first term is, 16x2 its coefficient is 16 .
The middle term is, +40x its coefficient is 40 .
The last term, "the constant", is +25
Step-1 : Multiply the coefficient of the first term by the constant 16 • 25 = 400
Step-2 : Find two factors of 400 whose sum equals the coefficient of the middle term, which is 40 .
| -400 | + | -1 | = | -401 | ||
| -200 | + | -2 | = | -202 | ||
| -100 | + | -4 | = | -104 | ||
| -80 | + | -5 | = | -85 | ||
| -50 | + | -8 | = | -58 | ||
| -40 | + | -10 | = | -50 | ||
| -25 | + | -16 | = | -41 | ||
| -20 | + | -20 | = | -40 | ||
| -16 | + | -25 | = | -41 | ||
| -10 | + | -40 | = | -50 | ||
| -8 | + | -50 | = | -58 | ||
| -5 | + | -80 | = | -85 | ||
| -4 | + | -100 | = | -104 | ||
| -2 | + | -200 | = | -202 | ||
| -1 | + | -400 | = | -401 | ||
| 1 | + | 400 | = | 401 | ||
| 2 | + | 200 | = | 202 | ||
| 4 | + | 100 | = | 104 | ||
| 5 | + | 80 | = | 85 | ||
| 8 | + | 50 | = | 58 | ||
| 10 | + | 40 | = | 50 | ||
| 16 | + | 25 | = | 41 | ||
| 20 | + | 20 | = | 40 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 20 and 20
16x2 + 20x + 20x + 25
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (4x+5)
Add up the last 2 terms, pulling out common factors :
5 • (4x+5)
Step-5 : Add up the four terms of step 4 :
(4x+5) • (4x+5)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (4x+5) by (4x+5)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (4x+5) and the exponents are :
1 , as (4x+5) is the same number as (4x+5)1
and 1 , as (4x+5) is the same number as (4x+5)1
The product is therefore, (4x+5)(1+1) = (4x+5)2
Final result :
(4x + 5)2
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