Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=x^{3}-9 x\), between \(x=-4\) and \(x=-2\)

Step-by-Step Solution

Answer

The polynomial has at least one zero between \( x = -4 \) and \( x = -2 \).

Step 1: Determine the function values at the interval's endpoints

Calculate the value of the function at the endpoints of the interval. Start with the lower endpoint, \( x = -4 \). Compute \( f(-4) = (-4)^3 - 9(-4) = -64 + 36 = -28 \). Next, compute the value at the upper endpoint \( x = -2 \). \( f(-2) = (-2)^3 - 9(-2) = -8 + 18 = 10 \).

Step 2: Apply the Intermediate Value Theorem (IVT)

Check if there is a sign change between \( f(-4) \) and \( f(-2) \), indicating the presence of a zero. Since \( f(-4) = -28 \) and \( f(-2) = 10 \), there is a sign change from negative to positive, meaning zero must be somewhere between \( x=-4 \) and \( x=-2 \).

Step 3: Confirm the existence of a zero

By the Intermediate Value Theorem, since \( f(x) \) is continuous on the interval \([-4, -2]\) and there is a sign change from negative to positive, \( f(x) \) must have at least one real zero in the interval. This confirms the existence of a zero between \( x = -4 \) and \( x = -2 \).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are algebraic expressions consisting of terms in the form of coefficients and variables raised to whole number powers. In simpler terms, they're expressions like \( x^3 - 9x \), where we see powers of \( x \) combined with numerical values. One of the main features of polynomial functions is their degree, which is the highest power of \( x \). The degree tells us about the function's behavior:

• A higher degree usually means the function has more curves.
• The degree also determines how many zeros (roots) the function might have, though some could be complex numbers not visible on the real number line.

Polynomial functions are continuous by nature, which is a crucial point for applying the Intermediate Value Theorem. They don't have breaks, jumps, or holes, making them perfect candidates for this theorem, which helps us find zeros within an interval. Understanding the structure and degree of a polynomial can help predict how it behaves across its domain.

Zeros of a Function: Finding the Solutions

The zeros of a function are the x-values where the function equals zero. For a polynomial like \( f(x) = x^3 - 9x \), finding the zeros means solving the equation \( x^3 - 9x = 0 \). This can often be done by factoring or using other algebraic techniques. Here's why zeros are important:

• Zeros are the points where the graph of the function crosses or touches the x-axis.
• In real-world terms, they could represent key changes or balances – like where profit equals cost.

For a continuous function, like a polynomial, the existence of zeros can often be confirmed using methods like the Intermediate Value Theorem, especially over a specified interval. Knowing how to locate these zeros gives deeper insight into the behavior of the function.

Continuous Functions and the Role They Play

A continuous function is one that can be drawn without lifting your pencil from the paper. This might sound simple, but it's a fundamental property that allows us to apply powerful tools like the Intermediate Value Theorem. Such functions don't have breaks, jumps, or holes which makes them predictable.

• In terms of polynomials, this means you can expect a smooth curve.
• Continuity guarantees that any change in y-values happens smoothly over the x-values.

This is especially useful with the Intermediate Value Theorem because it states that for a continuous function like a polynomial on a closed interval, if the function values at the endpoints are of opposite signs, then there must be at least one zero within that interval. This property helps us confirm the existence of zeros without explicitly solving for them.