Problem 35 - Algebra and Trigonometry

Question

In Exercises \(33-40,\) use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=2 x^{4}-4 x^{2}+1 ; \text { between }-1 \text { and } 0$$

Step-by-Step Solution

Answer

Using Intermediate Value Theorem, it is shown that the polynomial \(2x^4 -4x^2 +1\) has a real zero between -1 and 0.

Step 1: Compute \(f(-1)\)

The first step is to find the value of function \(f(x)\) at \(x=-1\). This can be done by substituting \(x=-1\) into the polynomial, yielding \(f(-1) = 2(-1)^4 -4(-1)^2 +1 = 2-4+1=-1\).

Step 2: Compute \(f(0)\)

Next, find the value of function \(f(x)\) at \(x=0\). This can be done by substituting \(x=0\) into the polynomial, which results in \(f(0) = 2(0)^4 -4(0)^2 +1 = 1\).

Step 3: Apply Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous in the interval [a, b] and \(f(a)\) and \(f(b)\) are the function values at endpoints, then it takes on any value between \(f(a)\) and \(f(b)\) at some point in the interval. Here, \(f(a)=-1\) and \(f(b)=1\). Since \(0\) is a value between \(-1\) and \(1\), there must be a value in the interval \([-1,0]\) for which the polynomial equals zero. This shows that the polynomial \(2x^4 -4x^2 +1\) has a real zero between -1 and 0.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables multiplied by coefficients. The general form can be expressed as:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]where:

• \(a_n, a_{n-1}, ..., a_1, a_0\) are coefficients.
• \(n\) is a non-negative integer representing the highest power of \(x\).
• \(a_n eq 0\), meaning the leading coefficient is not zero.

Polynomial functions exhibit smooth, continuous curves when graphed. This smoothness makes them ideal for analyzing changes and trends. They can range from simple linear functions to complex fourth-degree (quartic) functions as seen in the problem: \(2x^4 - 4x^2 + 1\). Understanding these expressions is crucial as they form the basis for more complex calculus applications.

Real Zeros

Real zeros of a polynomial are the values of \(x\) at which the function equals zero. They are also the points where the graph of the polynomial crosses the x-axis. Finding these zeros is essential as they give insights into the function's behavior and where it shifts sign or changes direction.
To determine real zeros using the Intermediate Value Theorem, you check if the function changes signs over an interval, indicating the presence of a zero. In our problem, since \(f(-1) = -1\) and \(f(0) = 1\), the function indeed changes sign, confirming the existence of a real zero between \(-1\) and \(0\). Real zeros are fundamental for solving polynomial equations and for graph manipulation.

Continuous Functions

A function is continuous if you can draw its graph without lifting your pen from the paper. More formally, a function \(f(x)\) is continuous over an interval if there are no breaks, jumps, or holes in that interval. Polynomial functions, like \(2x^4 - 4x^2 + 1\), are inherently continuous over all real numbers because their structure involves only additions, multiplications, and non-negative integer powers of x.
Continuity ensures that intermediate values between any two points within an interval are covered by the function. This property is vital when applying the Intermediate Value Theorem, as it guarantees that values between \(f(a)\) and \(f(b)\) are indeed taken by the function within the interval \([a, b]\). This continuity allows us to make reliable predictions about polynomial behavior.

Interval Arithmetic

Interval arithmetic involves analyzing function values over a specific range. When solving problems, such as finding zeros with the Intermediate Value Theorem, interval arithmetic helps assess changes across a span.
For instance, consider the interval between \(-1\) and \(0\). Calculating \(f(-1)\) gives \(-1\) and \(f(0)\) gives \(1\). Because the function changes from a negative value to a positive one, interval arithmetic suggests that the function passes through zero within this interval. This systematic approach is essential for determining real zeros efficiently and is widely used in numerical methods to narrow down roots or explore function behavior.