Solve: x^4−6x^3+4x^2+15x+4=0.

Verified step by step guidance

1. First, identify if there are any obvious roots by using the Rational Root Theorem, which suggests that any rational root, in the form of $\frac{p}{q}$, is a factor of the constant term (4) divided by a factor of the leading coefficient (1).
2. List the possible rational roots: $\pm 1,\pm 2,\pm 4$.
3. Use synthetic division or polynomial division to test these possible roots. Start with $x=1$ and continue testing until you find a root that results in a remainder of zero.
4. Once a root is found, factor the polynomial by dividing it by $x-\text{(found root)}$ to reduce the polynomial's degree.
5. Repeat the process with the reduced polynomial to find additional roots, if necessary, until the polynomial is fully factored into linear factors or irreducible quadratic factors.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the equation x^4−6x^3+4x^2+15x+4=0 is a polynomial of degree four, which indicates it can have up to four real roots. Understanding the behavior of polynomial functions, including their end behavior and turning points, is essential for solving polynomial equations.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of simpler polynomials or linear factors. This technique is crucial for solving polynomial equations, as it can simplify the process of finding roots. For the given equation, identifying possible rational roots or using synthetic division can help factor the polynomial, making it easier to solve for x.

The Rational Root Theorem

The Rational Root Theorem provides a method for identifying possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. Applying this theorem to the polynomial in the question can help narrow down the candidates for roots, facilitating the solving process.