In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=x³+2x²+5x+4

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^3+2x^2+5x+4$.
2. Apply Descartes's Rule of Signs to find the possible number of positive real zeros by counting the number of sign changes in the coefficients of $f(x)$. List the signs of the coefficients in order: +, +, +, +.
3. Since there are no sign changes in $f(x)$, conclude that there are 0 possible positive real zeros.
4. To find the possible number of negative real zeros, evaluate $f(-x)$ by substituting $-x$ into the function: $f(-x)\; =\; (-x)^3\; +\; 2(-x)^2\; +\; 5(-x)\; +\; 4$. Simplify the expression.
5. Count the number of sign changes in the coefficients of $f(-x)$ after simplification to determine the possible number of negative real zeros according to Descartes's Rule of Signs.

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a method used to determine the possible number of positive and negative real zeros of a polynomial function. It involves counting the number of sign changes in the coefficients of f(x) for positive zeros and in f(-x) for negative zeros. The actual number of positive or negative zeros is either equal to the number of sign changes or less than it by an even number.

Polynomial Functions and Their Zeros

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. The zeros of a polynomial are the values of x for which the function equals zero. Understanding the degree and coefficients helps in analyzing the behavior and possible roots of the polynomial.

Evaluating f(-x) to Find Negative Zeros

To apply Descartes's Rule of Signs for negative zeros, substitute -x into the polynomial to get f(-x). This changes the signs of terms with odd powers of x. Counting the sign changes in f(-x) reveals the possible number of negative real zeros, following the same rule as for positive zeros.