Problem 62 - Algebra 2

Question

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. $$ g(x)=2 x^{4}-x^{3}-3 x+7 $$

Step-by-Step Solution

Answer

2 positive real zeros or 0 positive, 0 negative real zeros, and 2 or 4 imaginary zeros.

Step 1: Determine Maximum Number of Zeros

The polynomial function \( g(x) = 2x^4 - x^3 - 3x + 7 \) is of degree 4. Therefore, by the Fundamental Theorem of Algebra, it can have at most 4 zeros (real or complex).

Step 2: Identify Positive Real Zeros Using Descartes' Rule of Signs

For positive real zeros, look at the sign changes in \( g(x) = 2x^4 - x^3 - 0x^2 - 3x + 7 \). The signs are: 1. From \( 2x^4 \) to \( -x^3 \) (positive to negative)2. From \( -3x \) to \( 7 \) (negative to positive)Thus, there are two sign changes, indicating 2 or 0 positive real zeros.

Step 3: Identify Negative Real Zeros Using Descartes' Rule of Signs

For negative real zeros, evaluate \( g(-x) = 2(-x)^4 - (-x)^3 - 3(-x) + 7 = 2x^4 + x^3 + 3x + 7 \). The signs are: There are no sign changes. Thus, there are 0 negative real zeros.

Step 4: Determine Number of Imaginary Zeros

Using the results from Steps 1, 2, and 3, remember that complex zeros (imaginary) occur in conjugate pairs. Since there are at most 4 zeros in total and at least 2 must be imaginary, possibilities for zero configurations are either 2 positive real and 2 complex zeros, or all 4 are complex if the positive possibilities don't occur.

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a central concept in understanding polynomial equations. It states that every non-zero polynomial equation of degree \( n \) will have exactly \( n \) roots, if you count each root based on its multiplicity.
This means a polynomial function with degree 4, like our function \( g(x) = 2x^4 - x^3 - 3x + 7 \), can have exactly 4 roots.
However, these roots can be real or complex numbers. The theorem ensures us that all roots fall within the realm of complex numbers, even if they are not real.
It does not, however, tell us the nature of these zeros (i.e., how many are real or complex), which is where other concepts, such as Descartes' Rule of Signs, come into play.

Descartes' Rule of Signs

Descartes' Rule of Signs offers a method to estimate the number of real roots of a polynomial equation by looking at the number of times the sign changes in the sequence of its coefficients. This is done for both positive and negative roots separately.
For positive real zeros, count the sign changes in \( g(x) = 2x^4 - x^3 - 0x^2 - 3x + 7 \).
In this case, we identify two sign changes:

• From \(2x^4\) (positive) to \(-x^3\) (negative)
• From \(-3x\) (negative) to \(7\) (positive)

Therefore, there could be 2 or 0 positive real zeros.
For negative real roots, evaluate \( g(-x) = 2x^4 + x^3 + 3x + 7 \).
Here, there are zero sign changes, which indicates there are 0 negative real zeros.

complex zeros

Complex zeros, commonly referred to as imaginary zeros when they involve imaginary numbers, always occur in conjugate pairs in polynomial equations with real coefficients. This means if \(a + bi\) is a zero of the polynomial, \(a - bi\) must also be a zero.
For our fourth-degree polynomial \(g(x) = 2x^4 - x^3 - 3x + 7\), we accounted for possibilities of 2 or 0 positive real zeros and 0 negative real zeros.
Given that complex roots must adhere to this conjugate pairing, if there are positive real zeros, the rest will be complex.
This leads us to two scenarios: if there are 2 positive real zeros, then there will be 2 complex zeros.
If none of the positive zeros manifest, then all 4 will be complex zeros, each forming pairs.

real zeros

Real zeros of a polynomial, as the name suggests, are the roots that are actual real numbers. These are points where the polynomial crosses or touches the x-axis of a graph.
In our polynomial \(g(x) = 2x^4 - x^3 - 3x + 7\), according to Descartes' Rule of Signs, there are 2 or 0 positive real zeros, and 0 negative real zeros.
This means our real zeros could purely be positive, or they might not exist at all if the real possibilities don't occur.
If real zeros do appear, they align with the number of sign changes for positive numbers. However, when considering the polynomial's total degree, any absence of real zeros emphasizes the presence of complex zeros instead.