Question

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ; \quad\) Irrational zeros: 0 (c) Rational zeros: 1; Irrational zeros: 2 (d) Rational zeros: 1; Irrational zeros: 0 $$f(x)=x^{3}-x$$

Short answer

None of the given options match with the number of rational (2) and the number of irrational zeros (1) of the function \(f(x)=x^{3}-x\).

Step-by-step solution

Apply Rational Root Theorem

The Rational Root Theorem gives us a potential list of rational zeros of a polynomial function. By applying the theorem to the polynomial \(f(x)=x^{3}-x\), we find that the potential rational zeros are \(±1, ±x\).

Test the Potential Zeros

Put each potential rational zero into the equation \(f(x)=x^{3}-x\) and see if it equates to zero. Testing \(±1\) gives \(f(1)=1^{3}-1=0\) and \(f(-1)=(-1)^{3}-(-1)=0\). However, putting \(±x\) in doesn't make sense as \(x\) is the variable of the function. So, there are 2 rational zeros, \(1\) and \(-1\).

Calculate the total number of zeros

In a cubic function, the total number of zeros (including both rational and irrational) can't exceed the degree of the function. Here, the degree of the function is 3. As we've found 2 rational zeros, the remaining number of zeros can't exceed 1.

Identify the number of irrational zeros

Subtract the number of rational zeros from the total number of zeros. This means the number of irrational zeros = 3 (total number of zeros) - 2 (rational zeros) = 1.

Match with the given options

Option (a) with rational zeros: 0 and irrational zeros: 1 doesn't match. Option (b) with rational zeros: 3 and irrational zeros: 0 also doesn't match. Option (c) with rational zeros: 1 and irrational zeros: 2, no match. But, Option (d) with rational zeros: 1 and irrational zeros: 0, also doesn't match as well. None of the options match with the calculated number of rational (2) and irrational zeros (1).

Key concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool when trying to find the rational zeros of a polynomial. It tells us that if a polynomial has a rational zero, it must be of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient. This theorem helps narrow down the possible rational zeros to a manageable list that we can test. When dealing with the cubic function \(f(x) = x^3 - x\), for example, the potential rational zeros we find are \( \pm 1 \) after applying this theorem. These numbers are derived because they are factors of the constant term - which is zero in this case, allowing simple calculations.

Once we have this list, it becomes much easier to determine the actual rational roots by plugging these numbers into the polynomial to see which will cause the polynomial to zero out. This direct method helps avoid testing every possible number, saving time and effort.

• The constant term in \(f(x) = x^3 - x\) is zero.
• The possible rational values that can make \(f(x) = 0\) are simply \(-1\) and \(1\).

Understanding this theorem offers a straightforward beginning point when solving chances of rationality within any polynomial.

Polynomial Degree

The degree of a polynomial is a crucial aspect to understand since it gives us information on how many roots - or zeros - the polynomial can have. Simply put, a polynomial's degree is the highest power of the variable present. For a cubic function like \(f(x) = x^3 - x\), the highest power of \(x\) is 3. This tells us the function is of degree 3.

Why is this important? Because the degree of the polynomial sets the maximum number of zeros it can have. In this case, a cubic polynomial will have exactly three zeros in total, counting multiple roots and complex ones as necessary. These zeros could be rational or irrational and knowing the degree lets us find all possible solutions. This understanding forms the basis for further analysis, especially when testing for real and rational roots using methods like the Rational Root Theorem.

• Degree reflects the maximum number of zeros.
• A polynomial of degree \(n\) has \(n\) total zeros.

This knowledge provides a framework for identifying zeros within polynomial equations.

Zeros of Polynomial

Zeros of a polynomial can be likened to the x-values where the polynomial equals zero. These zeros, also called roots, are essential in understanding the behavior of the polynomial function. There are two main types of zeros: rational and irrational.

Rational zeros are values that can be expressed as fractions or integers. On the other hand, irrational zeros are roots that cannot be precisely expressed as a simple fraction; they often include square roots or other complex irrational numbers. The polynomial \(f(x) = x^3 - x\) has both kinds.A step-by-step method is often employed to "test" whether a potential solution is indeed a zero. This involves substituting proposed zeros back into the polynomial and checking for a zero remainder.

• Rational zeros can be identified using the Rational Root Theorem.
• Irrational zeros might require other methods such as factoring or using the quadratic formula.

Once all zeros are determined, comparing them against given options can highlight any discrepancies or misinterpretations in exercises or examples, ensuring a thorough understanding and correct identification of all zeros of the polynomial.