Step by Step Solution
Step 1 :
Equation at the end of step 1 :
((((x4)-(32•(x3)))+(27•3x2))-2048x)+4096Step 2 :
Equation at the end of step 2 :
((((x4) - 25x3) + (27•3x2)) - 2048x) + 4096
Step 3 :
Polynomial Roots Calculator :
3.1 Find roots (zeroes) of : F(x) = x4-32x3+384x2-2048x+4096
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x 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 1 and the Trailing Constant is 4096.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 ,256 ,512 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 6561.00 | ||||||
| -2 | 1 | -2.00 | 10000.00 | ||||||
| -4 | 1 | -4.00 | 20736.00 | ||||||
| -8 | 1 | -8.00 | 65536.00 | ||||||
| -16 | 1 | -16.00 | 331776.00 | ||||||
| -32 | 1 | -32.00 | 2560000.00 | ||||||
| -64 | 1 | -64.00 | 26873856.00 | ||||||
| -128 | 1 | -128.00 | 342102016.00 | ||||||
| -256 | 1 | -256.00 | 4857532416.00 | ||||||
| -512 | 1 | -512.00 | 73116160000.00 | ||||||
| 1 | 1 | 1.00 | 2401.00 | ||||||
| 2 | 1 | 2.00 | 1296.00 | ||||||
| 4 | 1 | 4.00 | 256.00 | ||||||
| 8 | 1 | 8.00 | 0.00 | x-8 | |||||
| 16 | 1 | 16.00 | 4096.00 | ||||||
| 32 | 1 | 32.00 | 331776.00 | ||||||
| 64 | 1 | 64.00 | 9834496.00 | ||||||
| 128 | 1 | 128.00 | 207360000.00 | ||||||
| 256 | 1 | 256.00 | 3782742016.00 | ||||||
| 512 | 1 | 512.00 | 64524128256.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x4-32x3+384x2-2048x+4096
can be divided with x-8
Polynomial Long Division :
3.2 Polynomial Long Division
Dividing : x4-32x3+384x2-2048x+4096
("Dividend")
By : x-8 ("Divisor")
| dividend | x4 | - | 32x3 | + | 384x2 | - | 2048x | + | 4096 | ||
| - divisor | * x3 | x4 | - | 8x3 | |||||||
| remainder | - | 24x3 | + | 384x2 | - | 2048x | + | 4096 | |||
| - divisor | * -24x2 | - | 24x3 | + | 192x2 | ||||||
| remainder | 192x2 | - | 2048x | + | 4096 | ||||||
| - divisor | * 192x1 | 192x2 | - | 1536x | |||||||
| remainder | - | 512x | + | 4096 | |||||||
| - divisor | * -512x0 | - | 512x | + | 4096 | ||||||
| remainder | 0 |
Quotient : x3-24x2+192x-512 Remainder: 0
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = x3-24x2+192x-512
See theory in step 3.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is -512.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32 ,64 ,128 ,256 ,512
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -729.00 | ||||||
| -2 | 1 | -2.00 | -1000.00 | ||||||
| -4 | 1 | -4.00 | -1728.00 | ||||||
| -8 | 1 | -8.00 | -4096.00 | ||||||
| -16 | 1 | -16.00 | -13824.00 | ||||||
| -32 | 1 | -32.00 | -64000.00 | ||||||
| -64 | 1 | -64.00 | -373248.00 | ||||||
| -128 | 1 | -128.00 | -2515456.00 | ||||||
| -256 | 1 | -256.00 | -18399744.00 | ||||||
| -512 | 1 | -512.00 | -140608000.00 | ||||||
| 1 | 1 | 1.00 | -343.00 | ||||||
| 2 | 1 | 2.00 | -216.00 | ||||||
| 4 | 1 | 4.00 | -64.00 | ||||||
| 8 | 1 | 8.00 | 0.00 | x-8 | |||||
| 16 | 1 | 16.00 | 512.00 | ||||||
| 32 | 1 | 32.00 | 13824.00 | ||||||
| 64 | 1 | 64.00 | 175616.00 | ||||||
| 128 | 1 | 128.00 | 1728000.00 | ||||||
| 256 | 1 | 256.00 | 15252992.00 | ||||||
| 512 | 1 | 512.00 | 128024064.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-24x2+192x-512
can be divided with x-8
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : x3-24x2+192x-512
("Dividend")
By : x-8 ("Divisor")
| dividend | x3 | - | 24x2 | + | 192x | - | 512 | ||
| - divisor | * x2 | x3 | - | 8x2 | |||||
| remainder | - | 16x2 | + | 192x | - | 512 | |||
| - divisor | * -16x1 | - | 16x2 | + | 128x | ||||
| remainder | 64x | - | 512 | ||||||
| - divisor | * 64x0 | 64x | - | 512 | |||||
| remainder | 0 |
Quotient : x2-16x+64 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring x2-16x+64
The first term is, x2 its coefficient is 1 .
The middle term is, -16x its coefficient is -16 .
The last term, "the constant", is +64
Step-1 : Multiply the coefficient of the first term by the constant 1 • 64 = 64
Step-2 : Find two factors of 64 whose sum equals the coefficient of the middle term, which is -16 .
| -64 | + | -1 | = | -65 | ||
| -32 | + | -2 | = | -34 | ||
| -16 | + | -4 | = | -20 | ||
| -8 | + | -8 | = | -16 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and -8
x2 - 8x - 8x - 64
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-8)
Add up the last 2 terms, pulling out common factors :
8 • (x-8)
Step-5 : Add up the four terms of step 4 :
(x-8) • (x-8)
Which is the desired factorization
Multiplying Exponential Expressions :
3.6 Multiply (x-8) by (x-8)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-8) and the exponents are :
1 , as (x-8) is the same number as (x-8)1
and 1 , as (x-8) is the same number as (x-8)1
The product is therefore, (x-8)(1+1) = (x-8)2
Multiplying Exponential Expressions :
3.7 Multiply (x-8)2 by (x-8)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-8) and the exponents are :
2
and 1 , as (x-8) is the same number as (x-8)1
The product is therefore, (x-8)(2+1) = (x-8)3
Multiplying Exponential Expressions :
3.8 Multiply (x-8)3 by (x-8)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-8) and the exponents are :
3
and 1 , as (x-8) is the same number as (x-8)1
The product is therefore, (x-8)(3+1) = (x-8)4
Final result :
(x - 8)4
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