Use Descartes' Rule of Signs to explain why 2x^4 + 6x^2 + 8 = 0 has no real roots.

Verified step by step guidance

1. Identify the polynomial: $2x^4 + 6x^2 + 8$.
2. Count the number of sign changes in the polynomial $2x^4 + 6x^2 + 8$. Since all coefficients are positive, there are 0 sign changes.
3. According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less by an even number. Here, it is 0.
4. To find the number of negative real roots, substitute $x$ with $-x$ in the polynomial, resulting in $2(-x)^4 + 6(-x)^2 + 8 = 2x^4 + 6x^2 + 8$. Again, all coefficients are positive, so there are 0 sign changes.
5. Thus, according to Descartes' Rule of Signs, there are 0 negative real roots as well. Therefore, the polynomial has no real roots.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a mathematical theorem that provides a way to determine the number of positive and negative real roots of a polynomial function based on the number of sign changes in its coefficients. For positive roots, you count the sign changes in the polynomial as it is written. For negative roots, you evaluate the polynomial at -x and count the sign changes in that expression.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of the variable. In the case of the polynomial 2x^4 + 6x^2 + 8, the highest degree is 4, indicating it is a quartic polynomial, which can have up to four roots, real or complex.

Real Roots

Real roots of a polynomial are the values of the variable that make the polynomial equal to zero and are found on the real number line. A polynomial can have real roots, complex roots, or both. In the context of the given polynomial, analyzing the coefficients and applying Descartes' Rule of Signs reveals that there are no sign changes, indicating that there are no positive or negative real roots.