Problem 11 - Precalculus

Question

Determine whether the given value is a zero of the function. $$f(t)=1+2 t+t^{3}-t^{5} ; t=2$$

Step-by-Step Solution

Answer

The given value \( t = 2 \) is not a zero of the function.

Step 1: Substitute the Value

First, we need to check if the given value \( t = 2 \) is a zero of the function \( f(t) = 1 + 2t + t^3 - t^5 \). To do this, we substitute \( t = 2 \) into the function: \( f(2) = 1 + 2(2) + (2)^3 - (2)^5 \).

Step 2: Calculate Exponents

Now, calculate the exponents: \( (2)^3 = 8 \) and \( (2)^5 = 32 \). Substitute these values back into the expression: \( f(2) = 1 + 4 + 8 - 32 \).

Step 3: Simplify the Expression

Add and subtract the numbers in the expression: \( 1 + 4 = 5 \), then \( 5 + 8 = 13 \), and finally, \( 13 - 32 = -19 \).

Step 4: Determine if it's a Zero

Since \( f(2) = -19 \) and not \( 0 \), \( t = 2 \) is not a zero of the function.

Key Concepts

Polynomial function evaluation

Polynomials are mathematical expressions involving a collection of terms, formed by variables and coefficients. Evaluating these functions means calculating their output when certain values replace the variable. In our exercise, we're working with a polynomial function:
\[ f(t) = 1 + 2t + t^3 - t^5 \]
To determine if a specific value, such as \( t=2 \), is a zero of the function, we have to substitute \( t \) with 2 and solve the expression. Evaluating the polynomial helps us pinpoint if the output equals zero, which would indicate that the value is indeed a zero of the function.

• Check each term in the polynomial, ensuring you apply the operations correctly.
• Perform operations like addition, subtraction, and multiplication.
• Focus on evaluating terms separately and carefully to avoid errors.

Understanding polynomial evaluation is pivotal because it anchors much of algebra and calculus.

Exponentiation

Exponentiation is the process of raising numbers to powers. It's an essential operation in algebra, used to simplify expressions and solve equations. For the exercise here, exponentiation helps evaluate parts of the polynomial function:

• \( (2)^3 = 2 \times 2 \times 2 = 8 \)
• \( (2)^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)

Exponents indicate how many times a number (the base) is multiplied by itself. When dealing with polynomial functions, exponentiation represents the complexity of terms as they contain variables raised to different powers. Understanding this concept allows us to evaluate polynomial terms accurately.

Function substitution

Substitution in mathematics allows you to replace variables with actual numbers within an expression. This step is crucial in the process of determining zeros of functions. In our example:

• Replace \( t \) with 2 in the function \( f(t) = 1 + 2t + t^3 - t^5 \).
• Calculate the resulting expression.

By substituting \( t=2 \) into \( f(t) \), we substitute all occurrences of \( t \) with 2, generating a pure numerical expression to evaluate. It simplifies the task of testing if a number is a function's zero or not. Function substitution is relevant because it transforms abstract algebraic expressions into concrete evaluations—laying down the foundation for solving real-world problems.

Step-by-step problem solving

Taking problems one step at a time is vital for accuracy, clarity, and understanding, especially in mathematics. Breaking down a problem methodically:

• Start with substitution to simplify the expression.
• Proceed with the calculation of exponents.
• Simplify the expression through addition and subtraction.
• Conclude by checking if the evaluated expression equals zero.

Following these steps in sequence enables us to examine each part of a problem thoroughly. This approach aids students in verifying each step independently and decreases the chances of overlooking crucial details or making errors. Applying step-by-step problem-solving techniques empowers students, making complex problems more approachable.