Solution - Simplification or other simple results
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(32x2 + 24x) + 16
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 9x2+24x+16
The first term is, 9x2 its coefficient is 9 .
The middle term is, +24x its coefficient is 24 .
The last term, "the constant", is +16
Step-1 : Multiply the coefficient of the first term by the constant 9 • 16 = 144
Step-2 : Find two factors of 144 whose sum equals the coefficient of the middle term, which is 24 .
| -144 | + | -1 | = | -145 | ||
| -72 | + | -2 | = | -74 | ||
| -48 | + | -3 | = | -51 | ||
| -36 | + | -4 | = | -40 | ||
| -24 | + | -6 | = | -30 | ||
| -18 | + | -8 | = | -26 | ||
| -16 | + | -9 | = | -25 | ||
| -12 | + | -12 | = | -24 | ||
| -9 | + | -16 | = | -25 | ||
| -8 | + | -18 | = | -26 | ||
| -6 | + | -24 | = | -30 | ||
| -4 | + | -36 | = | -40 | ||
| -3 | + | -48 | = | -51 | ||
| -2 | + | -72 | = | -74 | ||
| -1 | + | -144 | = | -145 | ||
| 1 | + | 144 | = | 145 | ||
| 2 | + | 72 | = | 74 | ||
| 3 | + | 48 | = | 51 | ||
| 4 | + | 36 | = | 40 | ||
| 6 | + | 24 | = | 30 | ||
| 8 | + | 18 | = | 26 | ||
| 9 | + | 16 | = | 25 | ||
| 12 | + | 12 | = | 24 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 12 and 12
9x2 + 12x + 12x + 16
Step-4 : Add up the first 2 terms, pulling out like factors :
3x • (3x+4)
Add up the last 2 terms, pulling out common factors :
4 • (3x+4)
Step-5 : Add up the four terms of step 4 :
(3x+4) • (3x+4)
Which is the desired factorization
Multiplying Exponential Expressions :
2.2 Multiply (3x+4) by (3x+4)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (3x+4) and the exponents are :
1 , as (3x+4) is the same number as (3x+4)1
and 1 , as (3x+4) is the same number as (3x+4)1
The product is therefore, (3x+4)(1+1) = (3x+4)2
Final result :
(3x + 4)2
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