Problem 10 - Precalculus

Question

Determine whether the given value is a zero of the function. $$F(x)=-2 x^{5}+3 x^{4}+8 x^{3} ; x=0$$

Step-by-Step Solution

Answer

Yes, \( x = 0 \) is a zero of the function.

Step 1: Understand the Definition of a Zero

A value \( x = a \) is a zero of a function \( F(x) \) if substituting \( a \) into the function results in \( F(a) = 0 \). In other words, the function's value is zero at that point.

Step 2: Substitute the Given Value into the Function

Substitute the given \( x = 0 \) into the function \( F(x) = -2x^5 + 3x^4 + 8x^3 \). This involves replacing every occurrence of \( x \) with \( 0 \).

Step 3: Simplify the Expression

Calculate \( F(0) = -2(0)^5 + 3(0)^4 + 8(0)^3 = -2 \, \cdot \, 0 + 3 \, \cdot \, 0 + 8 \, \cdot \, 0 = 0 \).

Step 4: Conclusion

Since \( F(0) = 0 \), the given value \( x = 0 \) is indeed a zero of the function \( F(x) \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number exponents. These expressions add up several such terms. In a polynomial, each term consists of a coefficient multiplied by a power of the variable. The general form of a polynomial function in a single variable, such as \( x \), is written as:

• \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

Here, each \( a \) represents a coefficient, and \( n \) indicates the degree of the polynomial, which is the highest exponent observed. The characteristics of polynomial functions include:

• A continuous and smooth graph.
• Defined for all real numbers \( x \).
• Operations like adding, subtracting, and multiplying polynomials result in another polynomial.

Understanding the structure of polynomial functions helps when determining features like zeros, which are essential in analyzing and solving equations involving polynomials.

Evaluating Functions

Evaluating a function means finding the value of the function for a specific input. This involves substituting the given value for the variable in the function. Let's consider the function from the exercise, \( F(x) = -2x^5 + 3x^4 + 8x^3 \). If you need to find \( F(0) \), follow these steps:

• Replace every occurrence of \( x \) with the number zero.
• Simplify each term: since any power of zero is zero itself, \, \( 0^n = 0 \), each term becomes zero.
• Finally, add up these results.

Evaluating a function allows us to understand the behavior of the function at particular points and is particularly useful in verifying whether a certain point is a zero of the function. This is done by checking if the function equals zero for the specific input.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They represent values and operations that can be carried out to find a specific result. These expressions form the backbone of algebra and other mathematical calculations. In algebraic expressions, you will find:

• Variables that serve as placeholders for unknown numbers.
• Coefficients that are the numerical factors of terms involving variables.
• Operators such as addition, subtraction, multiplication, and division.

Consider the expression in the exercise: \( -2x^5 + 3x^4 + 8x^3 \). Here:

• The variable is \( x \).
• The terms are \( -2x^5, \; 3x^4, \; \text{and} \; 8x^3 \).
• Coefficients are \( -2, \; 3, \; \text{and} \; 8 \).
• Operations involved are addition and multiplication.

Understanding how to manipulate these elements allows one to simplify expressions effectively and solve equations by locating zeros, factoring, and more.