Solve: x⁴+2x³−x²−4x−2=0

Verified step by step guidance

1. Start by examining the polynomial equation $x^4+\; 2x^3-x^2-\; 4x\; -\; 2\; =\; 0$ to see if it can be factored by grouping or by using substitution.
2. Try to group terms to factor: group the first two terms and the last three terms separately, like this: ($x^4+\; 2x^3$) + ($-x^2-\; 4x\; -\; 2$). Then factor out the greatest common factor (GCF) from each group.
3. 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.
4. 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: $x\; =\; \backslash (\backslash frac\{-B\; \backslash pm\; \backslash sqrt\{B^2\; -\; 4AC\}\backslash )\}\{2A\}$, where A, B, and C are the coefficients of the quadratic.
5. Write down all the solutions obtained from the quadratic equations. These solutions are the roots of the original quartic equation.

Key Concepts

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.

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.

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.