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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 81
Capitolo 6, Problema 81

Solve: x4+2x3−x2−4x−2=0

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Start by examining the polynomial equation x4 + 2x3 - x2 - 4x - 2 = 0 to see if it can be factored by grouping or by using substitution.
Try to group terms to factor: group the first two terms and the last three terms separately, like this: (x4 + 2x3) + (- x2 - 4x - 2). Then factor out the greatest common factor (GCF) from each group.
After factoring by grouping, check if the resulting expression can be factored further into the product of two quadratic polynomials, such as (x^2 + ax + b)(x^2 + cx + d) = 0. Set up equations by expanding and matching coefficients to find the values of a, b, c, and d.
Once the polynomial is factored into two quadratics, set each quadratic equal to zero: x^2 + ax + b = 0 and x^2 + cx + d = 0. Solve each quadratic equation using the quadratic formula: x = \(\frac{-B \pm \sqrt{B^2 - 4AC}\)}{2A}, where A, B, and C are the coefficients of the quadratic.
Write down all the solutions obtained from the quadratic equations. These solutions are the roots of the original quartic equation.

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Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents and coefficients. Solving these equations means finding all values of the variable that make the equation true. Understanding the degree of the polynomial helps determine the maximum number of possible solutions.
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Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials. This technique is essential for solving polynomial equations because setting each factor equal to zero helps find the roots. Common methods include grouping, synthetic division, and recognizing special products.
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Rational Root Theorem and Root Testing

The Rational Root Theorem provides possible rational roots based on factors of the constant and leading coefficients. Testing these candidates by substitution or synthetic division helps identify actual roots, simplifying the polynomial and making it easier to solve completely.
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Exercises 86–88 will help you prepare for the material covered in the first section of the next chapter. a. Does (4, −1) satisfy x + 2y = 2? b. Does (4, -1) satisfy x- 2y= 6?

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Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

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Solve the systems in Exercises 79–80.

{logy x=3logy (4x)=5\(\left\)\{\(\begin{array}{l}\)\(\log\)_{y}\(\text{ x=3}\)\\ \(\log\)_{y}\(\text{ (4x)=5}\[\end{array}\]\right\).

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Solve the systems in Exercises 79–80.

{log x2=y+3log x=y−1\(\left\)\{\(\begin{array}{l}\[\text{log }\)x^2=y+3\\ \(\text{log }\)x^{}=y-1\(\end{array}\]\right\).


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