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This solution deals with approximation. Step by Step SolutionStep by step solution :Step 1 :Polynomial Roots Calculator : 1.1 Find roots (zeroes) of : F(x) = x4-2x-1 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 -1. The factor(s) are: of the Leading Coefficient : 1
Equation at the end of step 1 : x4 - 2x - 1 = 0
Step 2 :Quartic Equations :2.1 Solve x4-2x-1 = 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 x=-20 and x=+20 Found change of sign between x= -1.00 and x= 0.00 Approximating a root using the Bisection Method :We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation). The function is F(x) = x4 - 2x - 1 At x= 0.00 F(x) is equal to -1.00 Intuitively we feel, and justly so, that since F(x) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(x) is zero Procedure : (2) Find a point 'Right' where F(Right) > 0 (3) Compute 'Middle' the middle point of the interval [Left,Right] (4) Calculate Value = F(Middle) (5) If Value is close enough to zero goto Step (7) Else : (6) Loop back to Step (3) (7) Done!! The approximation found is Middle Follow Middle movements to understand how it works : Left Value(Left) Right Value(Right) 0.000000000 -1.000000000 -1.000000000 2.000000000 0.000000000 -1.000000000 -1.000000000 2.000000000 0.000000000 -1.000000000 -0.500000000 0.062500000 -0.250000000 -0.496093750 -0.500000000 0.062500000 -0.375000000 -0.230224609 -0.500000000 0.062500000 -0.437500000 -0.088363647 -0.500000000 0.062500000 -0.468750000 -0.014220238 -0.500000000 0.062500000 -0.468750000 -0.014220238 -0.484375000 0.023796141 -0.468750000 -0.014220238 -0.476562500 0.004704777 -0.472656250 -0.004778184 -0.476562500 0.004704777 -0.474609375 -0.000041859 -0.476562500 0.004704777 -0.474609375 -0.000041859 -0.475585938 0.002330165 -0.474609375 -0.000041859 -0.475097656 0.001143830 -0.474609375 -0.000041859 -0.474853516 0.000550905 -0.474609375 -0.000041859 -0.474731445 0.000254503 -0.474609375 -0.000041859 -0.474670410 0.000106317 -0.474609375 -0.000041859 -0.474639893 0.000032228 -0.474624634 -0.000004816 -0.474639893 0.000032228 -0.474624634 -0.000004816 -0.474632263 0.000013706
Next Middle will get us close enough to zero: F( -0.474626541 ) is -0.000000186 The desired approximation of the solution is: x ≓ -0.474626541 Note, ≓ is the approximation symbol One solution was found :x ≓ -0.474626541 Why learn thisTerms and topicsRelated linksHow do you find solution sets?To find the solution set from the replacement set, plug in each value from the replacement set and evaluate both sides of the equation. If the two sides are equal, the equation is true and thus the value is a solution. Example 1: Find the solution set of 11 - 5w = 1 from the replacement set {0, 2, 4}.
How do you solve for x?How Do You Solve for x? To solve for x, bring the variable to one side, and bring all the remaining values to the other side by applying arithmetic operations on both sides of the equation. Simplify the values to find the result.
What is the solution set to the inequality 5 x 2 )( x 4 0?Summary: The solution set to the inequality 5(x - 2)(x + 4) > 0 is (- ∞, -4) U (2, ∞).
What is the solution set of the inequality x 2 5?(−7,3)
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