Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁴+x³-x²+3; no real zero less than -2

Verified step by step guidance

1. First, understand the problem: we need to show that any real zero \( x \) of the polynomial function \( f(x) = x^4 + x^3 - x^2 + 3 \) is not less than \( -2 \). In other words, if \( f(x) = 0 \), then \( x \geq -2 \).
2. Evaluate the polynomial at \( x = -2 \) to check the value of \( f(-2) \). This helps determine if \( -2 \) is a root or if the function changes sign around this point. Write \( f(-2) = (-2)^4 + (-2)^3 - (-2)^2 + 3 \).
3. Analyze the behavior of \( f(x) \) for values less than \( -2 \). For example, pick a test value such as \( x = -3 \) and compute \( f(-3) \) to see if the function is positive or negative there. This helps identify if the function crosses the x-axis to the left of \( -2 \).
4. Use the Intermediate Value Theorem: if \( f(x) \) does not change sign between \( -10 \) and \( -2 \) (or any interval to the left of \( -2 \)), then there are no real zeros less than \( -2 \).
5. Optionally, analyze the derivative \( f'(x) \) to understand the shape and monotonicity of the polynomial on the interval \( (-1\text{, } -2) \), which can support the conclusion about the location of zeros.

Key Concepts

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x for which the function equals zero. Understanding how to find and interpret these zeros is essential for analyzing the behavior of the function.

Evaluating Polynomial Values at Specific Points

Evaluating a polynomial at a given point involves substituting the value into the function and calculating the result. This helps determine whether the function crosses the x-axis at or near that point, which is useful for verifying conditions about the location of zeros.

Using Inequalities to Bound Zeros

Inequalities can be used to establish intervals where zeros of a polynomial may or may not exist. By testing values and applying the Intermediate Value Theorem or sign analysis, one can show that no real zeros lie below or above a certain number, as required in the problem.