In Problems 57–68, solve each equation in the real number system. 2 x^4-19 x^3+57 x^2-64 x+20=0

Name: Problem 67, Chapter 5: Algebra and Trigonometry
Uploaded: 2020-06-18T20:08:54-08:00
Duration: 2 min 38 s
Channel: Karly Williams
Description: In Problems 57–68, solve each equation in the real number system. 2 x^4-19 x^3+57 x^2-64 x+20=0

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In Problems 57–68, solve each equation in the real number...

Key Concepts

Polynomial Equations

Polynomial equations are algebraic expressions set equal to zero, composed of terms involving powers of the variable. They can be solved by finding the values of the variable that satisfy the equation. This process often involves techniques that reduce higher degree polynomials into simpler, more manageable forms.

Degree of a Polynomial

The degree of a polynomial is the highest exponent on the variable in the expression. It provides vital information about the behavior, number of possible roots, and overall structure of the equation. In a fourth-degree polynomial, for example, there can be up to four roots in the complex number system, though some may be real.

Rational Root Theorem

The Rational Root Theorem is a useful tool for identifying possible rational solutions to a polynomial equation. It states that any rational solution, expressed as a fraction in simplest form, has a numerator that divides the constant term and a denominator that divides the leading coefficient. This theorem narrows down the candidates for testing as potential roots of the equation.

Synthetic Division

Synthetic division is a streamlined method of polynomial division, particularly useful when testing candidate roots provided by the Rational Root Theorem. It simplifies the division process and helps factor the polynomial by confirming a candidate as a root, thereby reducing the polynomial's degree.

Factorization

Factorization involves expressing a polynomial as a product of simpler polynomials. Breaking down a polynomial into factors often leads to lower degree equations that are easier to solve, such as linear or quadratic equations. This method is a foundational strategy when solving higher degree polynomial equations.

Quadratic Formula

The quadratic formula is a standard method for finding the roots of any quadratic equation. When reducing a higher degree polynomial by factorization leads to a quadratic component, this formula directly provides the solutions for that piece, ensuring that all possible roots are determined.

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