Step by Step Solution
Step by step solution :
Step 1 :
Polynomial Roots Calculator :
1.1 Find roots (zeroes) of : F(y) = y5+y-1028
Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y 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 -1028.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,257 ,514 ,1028
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -1030.00 | ||||||
| -2 | 1 | -2.00 | -1062.00 | ||||||
| -4 | 1 | -4.00 | -2056.00 | ||||||
| -257 | 1 | -257.00 | -1121154894342.00 | ||||||
| -514 | 1 | -514.00 | -35876956579366.00 | ||||||
| -1028 | 1 | -1028.00 | -1148062610492424.00 | ||||||
| 1 | 1 | 1.00 | -1026.00 | ||||||
| 2 | 1 | 2.00 | -994.00 | ||||||
| 4 | 1 | 4.00 | 0.00 | y-4 | |||||
| 257 | 1 | 257.00 | 1121154892286.00 | ||||||
| 514 | 1 | 514.00 | 35876956577310.00 | ||||||
| 1028 | 1 | 1028.00 | 1148062610490368.00 |
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
y5+y-1028
can be divided with y-4
Polynomial Long Division :
1.2 Polynomial Long Division
Dividing : y5+y-1028
("Dividend")
By : y-4 ("Divisor")
| dividend | y5 | + | y | - | 1028 | ||||||||
| - divisor | * y4 | y5 | - | 4y4 | |||||||||
| remainder | 4y4 | + | y | - | 1028 | ||||||||
| - divisor | * 4y3 | 4y4 | - | 16y3 | |||||||||
| remainder | 16y3 | + | y | - | 1028 | ||||||||
| - divisor | * 16y2 | 16y3 | - | 64y2 | |||||||||
| remainder | 64y2 | + | y | - | 1028 | ||||||||
| - divisor | * 64y1 | 64y2 | - | 256y | |||||||||
| remainder | 257y | - | 1028 | ||||||||||
| - divisor | * 257y0 | 257y | - | 1028 | |||||||||
| remainder | 0 |
Quotient : y4+4y3+16y2+64y+257 Remainder: 0
Polynomial Roots Calculator :
1.3 Find roots (zeroes) of : F(y) = y4+4y3+16y2+64y+257
See theory in step 1.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is 257.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,257
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 206.00 | ||||||
| -257 | 1 | -257.00 | 4295612622.00 | ||||||
| 1 | 1 | 1.00 | 342.00 | ||||||
| 257 | 1 | 257.00 | 4431442262.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 1 :
(y4 + 4y3 + 16y2 + 64y + 257) • (y - 4) = 0
Step 2 :
Theory - Roots of a product :
2.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Quartic Equations :
2.2 Solve y4+4y3+16y2+64y+257 = 0
In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.
Method of search: Calculate polynomial values for all integer points between y=-20 and y=+20
No interval at which a change of sign occures has been found. Consequently, Bisection Approximation can not be used. As this is a polynomial of an even degree it may not even have any real (as opposed to imaginary) roots
Solving a Single Variable Equation :
2.3 Solve : y-4 = 0
Add 4 to both sides of the equation :
y = 4
One solution was found :
y = 4How did we do?
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