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

Verified step by step guidance

1. Write down the polynomial: \$2x^4 + 6x^2 + 8 = 0$.
2. Identify the signs of the coefficients in the polynomial: all coefficients (2, 6, and 8) are positive.
3. Apply Descartes' Rule of Signs to the polynomial \(f(x)\) by counting the number of sign changes in the sequence of coefficients. Since all coefficients are positive, there are 0 sign changes.
4. According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less than that by an even number. Here, it means there are 0 positive real roots.
5. Next, apply Descartes' Rule of Signs to \(f(-x)\) by substituting \(-x\) into the polynomial and checking the signs of the coefficients. Since all powers of \(x\) are even, \(f(-x) = 2x^4 + 6x^2 + 8\) has the same coefficients and no sign changes, so there are 0 negative real roots as well.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a method to determine the possible number of positive and negative real roots of a polynomial by counting the sign changes in its coefficients. The number of positive real roots is either equal to the number of sign changes or less than it by an even number. To find negative roots, the rule is applied to the polynomial with x replaced by -x.

Polynomial Roots and Real Solutions

A root of a polynomial is a value of x that makes the polynomial equal to zero. Real roots are solutions that are real numbers, as opposed to complex or imaginary roots. Understanding the nature of roots helps in analyzing the behavior of polynomial equations and their graphs.

Sign Changes in Polynomial Coefficients

Sign changes occur when consecutive coefficients in a polynomial switch from positive to negative or vice versa. Counting these changes is essential in applying Descartes' Rule of Signs, as it directly relates to the maximum number of positive or negative real roots the polynomial can have.