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Chapter 3: Problem 93

Use the rational zeros theorem to list the possible rational zeros to the function \(f(x)=2 x^{3}-5 x^{2}+7 x-6\)

Short Answer

Expert verified
The possible rational zeros of the function are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Step by step solution

01

Identify the coefficients

Identify the leading coefficient (the coefficient of the term with the highest degree) and the constant term in the polynomial. Here, the leading coefficient is 2 (from the term \(2x^3\)), and the constant term is -6.
02

List factors of the constant term

List all the factors of the constant term (-6). The factors of -6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
03

List factors of the leading coefficient

List all the factors of the leading coefficient (2). The factors of 2 are \( \pm 1, \pm 2 \).
04

Form all possible fractions

Form all possible ratios of the factors of the constant term to the factors of the leading coefficient. This gives us the possible rational zeros. The possible rational zeros are given by \( \frac{\text{factors of } -6}{\text{factors of } 2} \), which are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \).
05

List the possible rational zeros

Combine all possible fractions from the previous step: \( \{ \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \} \). These are the possible rational zeros according to the rational zeros theorem.

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Key Concepts

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

polynomial functions
A polynomial function is simply a mathematical expression that sums multiple terms.
Each term includes a variable (like x) raised to a power and multiplied by a coefficient. For example, in the expression 2x^3 - 5x^2 + 7x - 6, the terms separately are 2x^3, -5x^2, 7x, and -6.
Poly stands for many, and nomial comes from term, so polynomial functions contain many terms.
They can be written as f(x) = an*x^n + an-1*x^(n-1) + ... + a1*x + a0. Here, n is a non-negative integer, and the ai's are coefficients.
Polynomials are widely used in different areas of mathematics and science because they can model a variety of phenomena.
leading coefficient
The leading coefficient is the coefficient of the highest degree term in a polynomial function.
In our exercise, the polynomial is given as f(x) = 2x^3 - 5x^2 + 7x - 6.
Here, the term with the highest degree (3) is 2x^3. Therefore, the leading coefficient is the number multiplying the highest power of x, which is 2 in this case.
The leading coefficient significantly influences the behavior of the polynomial function, especially its end behavior.
constant term
The constant term in a polynomial is the term that does not contain any variable. It is just a number standing alone.
In our example, f(x) = 2x^3 - 5x^2 + 7x - 6, the constant term is -6.
The constant term plays a vital role in the Rational Zeros Theorem, as it helps determine the list of possible rational zeros of the polynomial function.
factors of constants
To find the possible rational zeros of a polynomial, we need to look at the factors of the constant term.
In our exercise, the constant term is -6. The factors of -6 are the numbers that multiply together to give -6.
  • They are \(\text{ }\text{\textpm} 1, \text{\textpm} 2, \text{\textpm} 3, \text{\textpm} 6 \text{ }\text{\)
  • Likewise, we also consider the factors of the leading coefficient, which is 2. Its factors are \(\text{ }\text{\textpm} 1 \text{ and } \text{\textpm} 2 \text{ }\text{\).

By forming all possible ratios of these factors, we get the potential rational zeros of the polynomial.

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