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Chapter 12: Problem 11

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

Short Answer

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

Step by step solution

01

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: \( f(2) = 1 + 2(2) + (2)^3 - (2)^5 \).
02

Calculate Exponents

Now, calculate the exponents: \( (2)^3 = 8 \) and \( (2)^5 = 32 \). Substitute these values back into the expression: \( f(2) = 1 + 4 + 8 - 32 \).
03

Simplify the Expression

Add and subtract the numbers in the expression: \( 1 + 4 = 5 \), then \( 5 + 8 = 13 \), and finally, \( 13 - 32 = -19 \).
04

Determine if it's a Zero

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攍aying 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.

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Most popular questions from this chapter

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{2 x^{5}-11 x^{4}-4 x^{3}+53 x^{2}-24 x-5}{2 x^{3}+x^{2}-10 x-5}$$

In a note that appeared in The Two-Year College Mathematics Journal [vol. \(12(1981), \text { pp. } 334-336],\) Professors Warren Page and Leo Chosid explain how the process of testing for rational roots can be shortened. In essence, their result is as follows. Suppose that we have a polynomial with integer coefficients and we are testing for a possible root \(p / q .\) Then, if a noninteger is generated at any point in the synthetic division process, \(p / q\) cannot be a root of the polynomial. For example, suppose we want to know whether \(4 / 3\) is a root of \(6 x^{4}-10 x^{3}+2 x^{2}-9 x+8=0\) The first few steps of the synthetic division are as follows. $$ \begin{array}{rrrrrr} 4 / 3 & 6 & -10 & 2 & -9 & 8 \\ \hline & & 8 & -8 / 3 & & \\ \hline & 6 & -2 & & & \\ \hline \end{array} $$ since the noninteger \(-8 / 3\) has been generated in the synthetic division process, the process can be stopped; \(4 / 3\) is not a root of the polynomial. Use this idea to shorten your work in testing to see whether the numbers \(3 / 4,1 / 8\). and \(-3 / 2\) are roots of the equation $$ 8 x^{5}-5 x^{4}+3 x^{2}-2 x-6=0 $$

Use the rational roots theorem and the remainder theorem to determine the roots of the equation \(x^{3}+2 x^{2}-5 x-6=0 .\) (This is to verify a statement made in Example \(2 .\) )

You are given an improper rational expression. First, use long division to rewrite the expression in the form (polynomial) \(+\) (proper rational expression) Next, obtain the partial fraction decomposition for the proper rational expression. Finally, rewrite the given improper rational expression in the form (polynomial) \(+\) (partial fractions) $$\frac{x^{6}+2 x^{5}+5 x^{4}-x^{2}-2 x-4}{x^{4}-1}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{x+1}{x^{4}-16}$$
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