Explanation Given: The polynomial equation is x 3 + x 2 + 2 x − 4 = 0 . Rule used: Rational Solutions Test: “The rational solutions of the polynomial equation a n x n + a n − 1 x n − 1 + ⋅ ⋅ ⋅ + a 1 x + a 0 = 0 with integer coefficients must be of the form p q , where p is a factor of the constant term a 0 and q is a factor of a n , the leading coefficient”. Procedure used: Synthetic division method: “To divide a polynomial by ( x − a ) : 1. Arrange the coefficients in descending powers of x , with a 0 for any missing power. Place a from ( x − a ) to the left of the coefficients. 2. Bring down the first coefficient to the third line. Multiply the last number in the third line by a and write the product in the second line under the next term. 3. Add the last number in the second line to the number above it in the first line. Continue this process until all numbers in the first line are used. 4. The third line represents the coefficients of the quotient, with the last number the remainder. The quotient is a polynomial of degree one less than the dividend. If the remainder is 0, x − a is a factor of the polynomial, and the polynomial can be written as the product of the divisor x − a and the quotient.” Calculation: The given equation is x 3 + x 2 + 2 x − 4 = 0 . Here, the cubic equation has exactly 3 solutions. Suppose that any non-real solution occur in conjugate pair, and then the remaining one solution should be real. Thus, the above equation has at least one real solution. Consider the polynomial function from given equation as p ( x ) = x 3 + x 2 + 2 x − 4 . Form Rational solutions test, the zeros of the polynomial p ( x ) = x 3 + x 2 + 2 x − 4 are factors of 4 divided by factors of 1. That is, the possible rational solutions are ± 1 , ± 2 , and ± 4 . Thus, the viewing window of x -values from − 4 to 4 contains all the possible rational solutions. Draw the graph of the polynomial p ( x ) = x 3 + x 2 + 2 x − 4 as shown below in Figure 1, From Figure 1, it is observed that the polynomial p ( x ) crosses the x -axis at ( 1 , 0 ) . That is, the value x = 1 satisfies the equation p ( x ) = 0 . Thus, the solution of the equation p ( x ) = x 3 + x 2 + 2 x − 4 is x = 1 . This implies, x − 1 is factor of the polynomial p ( x ) = x 3 + x 2 + 2 x − 4

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Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Chapter 6, Problem 27RE

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To find: The exact solution to the equation x3+x2+2x−4=0 in the complex number system.

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Chapter 6, Problem 26RE

Chapter 6, Problem 28RE

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Chapter 6 Solutions

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Ch. 6.1 - Graph the function h(x) = 3x3 + 5x2 x 10 on the...Ch. 6.1 - Graph the function f(x) = 2x3 3x2 6x on the...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - For Exercises 1116, match the polynomial function...Ch. 6.1 - For Exercises 1720, use the equation of the...Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Sketch a graph of any cubic polynomial function...Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Daily Revenue The daily revenue in dollars from...Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Starbucks Using data from 2005 and projected to...Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Salaries The median salary for male workers, in...Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.2 - Find the cubic function that models the data in...Ch. 6.2 - Find the cubic function that is the best fit for...Ch. 6.2 - Prob. 3E Ch. 6.2 - Find the quartic function that is the best fit for...Ch. 6.2 - Prob. 5E Ch. 6.2 - a. 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Prob. 18E Ch. 6.6 - Prob. 19E Ch. 6.6 - Prob. 20E Ch. 6.6 - Prob. 21E Ch. 6.6 - Prob. 22E Ch. 6.6 - Prob. 23E Ch. 6.6 - Prob. 24E Ch. 6.6 - Prob. 25E Ch. 6.6 - Combine factoring with graphical and/or numerical...Ch. 6.6 - Prob. 27E Ch. 6.6 - 28. 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The exact solution to the equation x 3 + x 2 + 2 x − 4 = 0 in the complex number system.

Solution Summary: The author explains the rational solutions of the polynomial equation x3+X2 +2x-4=0 in the complex number system.

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To find: The exact solution to the equation x3+x2+2x−4=0 in the complex number system.

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Chapter 6 Solutions