Solve: $x^4-6x^3+4x^2+15x+4=0$.

Verified step by step guidance

1. Start by examining the polynomial equation $x^4-6x^3+4x^2+15x+4=0$ and consider possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term over factors of the leading coefficient.
2. List the possible rational roots by taking factors of the constant term 4 (±1, ±2, ±4) and dividing by factors of the leading coefficient 1 (±1), so possible roots are ±1, ±2, ±4.
3. Test each possible root by substituting into the polynomial or using synthetic division to check if it yields zero, indicating a root.
4. Once a root is found, use synthetic division or polynomial division to divide the original polynomial by the corresponding factor $(x\; -\; r)$, where $r$ is the root found, to reduce the polynomial to a cubic or quadratic.
5. Solve the reduced polynomial (cubic or quadratic) by factoring further, using the quadratic formula, or other methods to find the remaining roots.

Key Concepts

Polynomial Equations

A polynomial equation involves expressions with variables raised to whole-number exponents and coefficients. Solving such equations means finding all values of the variable that make the equation true. Understanding the degree of the polynomial helps determine the number of possible roots.

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials. This technique simplifies solving equations by setting each factor equal to zero. Recognizing patterns like grouping or special products can aid in factoring higher-degree polynomials.

Rational Root Theorem and Synthetic Division

The Rational Root Theorem helps identify possible rational roots of a polynomial by considering factors of the constant and leading coefficients. Synthetic division is a streamlined method to test these roots and divide polynomials, making it easier to factor and solve the equation.