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Chapter 4: Problem 76

Find only the rational zeros of the function. If there are none, state this. $$f(x)=x^{3}-x^{2}-4 x+3$$

Short Answer

Expert verified
There are no rational zeros.

Step by step solution

01

- State the Rational Root Theorem

The Rational Root Theorem states that any rational zero of a polynomial function, when written in the form of \ \(\frac{p}{q} \), is such that p is a factor of the constant term and q is a factor of the leading coefficient.
02

- Identify p and q

For the polynomial function \(f(x)=x^{3}-x^{2}-4x+3\), the constant term is 3 and the leading coefficient is 1. Therefore, p is a factor of 3 (\(\pm 1, \pm 3\)) and q is a factor of 1 (\(\pm 1\)).
03

- List all possible rational zeros

Using the factors of p and q, the possible rational zeros are \(\pm 1, \pm 3\).
04

- Test the possible rational zeros

Substitute each possible rational zero into the function \(f(x)\) to see if it equals 0:1. For \(x = 1\): \(f(1) = 1^{3} - 1^{2} - 4 (1) + 3 = 1 - 1 - 4 + 3 = -1\) (not a zero)2. For \(x = -1\): \(f(-1) = (-1)^{3} - (-1)^{2} - 4 (-1) + 3 = -1 - 1 + 4 + 3 = 5\) (not a zero)3. For \(x = 3\): \(f(3) = 3^{3} - 3^{2} - 4 (3) + 3 = 27 - 9 - 12 + 3 = 9\) (not a zero)4. For \ x = -3\: \(f(-3) = (-3)^{3} - (-3)^{2} - 4 (-3) + 3 = -27 - 9 + 12 + 3 = -21\) (not a zero)
05

- Conclude the result

Based on testing all the possible rational zeros, none of them resulted in \(f(x) = 0\). Hence, there are no rational zeros for the polynomial function \(f(x) = x^{3}-x^{2}-4x+3\).

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Key Concepts

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

Polynomial Function
A polynomial function is a mathematical expression that consists of variables raised to whole number exponents and coefficients. For example, in the polynomial function \(f(x)=x^{3}-x^{2}-4x+3\), we have terms like \(x^{3}\), \(-x^{2}\), \(-4x\), and \(3\).
  • The highest exponent, known as the degree of the polynomial, is three for this function.
  • Each term has a coefficient, which multiplies the variable.
  • The constant term is \(3\), which does not have a variable.
Polynomial functions can have various degrees, and each degree affects the function's shape and the number of possible zeros (where the function equals zero).Identifying these zeros is crucial as they help us understand key characteristics of the function.
Rational Zeros
Rational zeros are the values of \(x\) for which the polynomial function equals zero and can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers.The Rational Root Theorem helps in efficiently finding possible rational zeros of a polynomial.It states that if a polynomial has a rational zero, \(\frac{p}{q}\):

  • `p` is a factor of the constant term.
  • `q` is a factor of the leading coefficient.

    • For the polynomial \(f(x)=x^{3}-x^{2}-4x+3\):
    • The constant term is \(3\), so `p` can be \(\text{卤1, 卤3}\).
    • The leading coefficient is \(1\), so `q` can be \(\text{卤1}\).
    Combining these factors, the possible rational zeros are \(\text{卤1, 卤3}\).Identifying rational zeros can help factorize the polynomial or further analyze the function's behavior.
    Testing Possible Zeros
    Testing each possible rational zero involves substituting them into the function and checking if the result equals zero.Let's apply this to our polynomial \(f(x)=x^{3}-x^{2}-4x+3\) by testing \(\text{卤1}\) and \(\text{卤3}\):
    • For \(x=1\): \(f(1)=1^{3}-1^{2}-4(1)+3 = -1\) (not a zero)
    • For \(x=-1\): \(f(-1)=(-1)^{3}-(-1)^{2}-4(-1)+3=5\) (not a zero)
    • For \(x=3\): \(f(3)=3^{3}-3^{2}-4(3)+3=9\) (not a zero)
    • For \(x=-3\): \(f(-3)=(-3)^{3}-(-3)^{2}-4(-3)+3=-21\) (not a zero)
    After testing all possible rational zeros, none of them resulted in \(f(x)=0\), so we conclude there are no rational zeros for this polynomial.This step-by-step testing process is essential in verifying potential rational zeros derived from the Rational Root Theorem.

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