Solution - Simplification or other simple results
Other Ways to Solve
Simplification or other simple resultsStep by Step Solution
Step 1 :
Equation at the end of step 1 :
((49 • (p8)) + (2•3•7p4)) + 9Step 2 :
Equation at the end of step 2 :
(72p8 + (2•3•7p4)) + 9
Step 3 :
Trying to factor by splitting the middle term
3.1 Factoring 49p8+42p4+9
The first term is, 49p8 its coefficient is 49 .
The middle term is, +42p4 its coefficient is 42 .
The last term, "the constant", is +9
Step-1 : Multiply the coefficient of the first term by the constant 49 • 9 = 441
Step-2 : Find two factors of 441 whose sum equals the coefficient of the middle term, which is 42 .
| -441 | + | -1 | = | -442 | ||
| -147 | + | -3 | = | -150 | ||
| -63 | + | -7 | = | -70 | ||
| -49 | + | -9 | = | -58 | ||
| -21 | + | -21 | = | -42 | ||
| -9 | + | -49 | = | -58 | ||
| -7 | + | -63 | = | -70 | ||
| -3 | + | -147 | = | -150 | ||
| -1 | + | -441 | = | -442 | ||
| 1 | + | 441 | = | 442 | ||
| 3 | + | 147 | = | 150 | ||
| 7 | + | 63 | = | 70 | ||
| 9 | + | 49 | = | 58 | ||
| 21 | + | 21 | = | 42 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 21 and 21
49p8 + 21p4 + 21p4 + 9
Step-4 : Add up the first 2 terms, pulling out like factors :
7p4 • (7p4+3)
Add up the last 2 terms, pulling out common factors :
3 • (7p4+3)
Step-5 : Add up the four terms of step 4 :
(7p4+3) • (7p4+3)
Which is the desired factorization
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(p) = 7p4+3
Polynomial Roots Calculator is a set of methods aimed at finding values of p for which F(p)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers p which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 7 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,7
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 10.00 | ||||||
| -1 | 7 | -0.14 | 3.00 | ||||||
| -3 | 1 | -3.00 | 570.00 | ||||||
| -3 | 7 | -0.43 | 3.24 | ||||||
| 1 | 1 | 1.00 | 10.00 | ||||||
| 1 | 7 | 0.14 | 3.00 | ||||||
| 3 | 1 | 3.00 | 570.00 | ||||||
| 3 | 7 | 0.43 | 3.24 |
Polynomial Roots Calculator found no rational roots
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(p) = 7p4+3
See theory in step 3.2
In this case, the Leading Coefficient is 7 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,7
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 10.00 | ||||||
| -1 | 7 | -0.14 | 3.00 | ||||||
| -3 | 1 | -3.00 | 570.00 | ||||||
| -3 | 7 | -0.43 | 3.24 | ||||||
| 1 | 1 | 1.00 | 10.00 | ||||||
| 1 | 7 | 0.14 | 3.00 | ||||||
| 3 | 1 | 3.00 | 570.00 | ||||||
| 3 | 7 | 0.43 | 3.24 |
Polynomial Roots Calculator found no rational roots
Multiplying Exponential Expressions :
3.4 Multiply (7p4+3) by (7p4+3)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (7p4+3) and the exponents are :
1 , as (7p4+3) is the same number as (7p4+3)1
and 1 , as (7p4+3) is the same number as (7p4+3)1
The product is therefore, (7p4+3)(1+1) = (7p4+3)2
Final result :
(7p4 + 3)2
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