- Analyze the polynomial \(f(x) = 2x^4 - 7x^3 - 4x^2 + 9x + 2\) to find the number of positive real zeros by checking the signs of its coefficients. There are 2 sign changes, so there are 2 or 0 positive real zeros.
- Determine the polynomial \(f(-x) = 2x^4 + 7x^3 - 4x^2 - 9x + 2\) and check its sign changes. There are 2 sign changes, so there are 2 or 0 negative real zeros.
- (a) Possible number(s) of positive real zeros: \(\boxed{2, 0}\)
- (b) Possible number(s) of negative real zeros: \(\boxed{2, 0}\)
Explanation
- Understanding Descartes' Rule of Signs
To determine the possible number of positive and negative real zeros of the polynomial \(f(x) = 2x^4 - 7x^3 - 4x^2 + 9x + 2\), we apply Descartes' Rule of Signs. This rule states that the number of positive real zeros is equal to the number of sign changes in the coefficients of \(f(x)\), or less than this number by an even integer. Similarly, the number of negative real zeros is equal to the number of sign changes in the coefficients of \(f(-x)\), or less than this number by an even integer.
- Analyzing Positive Real Zeros
First, we list the coefficients of \(f(x) = 2x^4 - 7x^3 - 4x^2 + 9x + 2\):
- The coefficients are \(2, -7, -4, 9, 2\).
- Examining the sign changes:
- From \(2\) to \(-7\) is a sign change.
- From \(-7\) to \(-4\) is not a sign change.
- From \(-4\) to \(9\) is a sign change.
- From \(9\) to \(2\) is not a sign change.
There are \(2\) sign changes in total. According to Descartes' Rule, the possible number of positive real zeros is \(2\) or \(2 - 2 = 0\).
- Analyzing Negative Real Zeros
Next, we determine the coefficients of \(f(-x)\) by substituting \(-x\) for \(x\):
\(f(-x) = 2(-x)^4 - 7(-x)^3 - 4(-x)^2 + 9(-x) + 2\)
\(f(-x) = 2x^4 + 7x^3 - 4x^2 - 9x + 2\)
- The coefficients are \(2, 7, -4, -9, 2\).
- Examining the sign changes:
- From \(2\) to \(7\) is not a sign change.
- From \(7\) to \(-4\) is a sign change.
- From \(-4\) to \(-9\) is not a sign change.
- From \(-9\) to \(2\) is a sign change.
There are \(2\) sign changes in total. According to Descartes' Rule, the possible number of negative real zeros is \(2\) or \(2 - 2 = 0\).
- Conclusion
Based on our analysis of the sign changes for both \(f(x)\) and \(f(-x)\), we conclude that the possible numbers of real zeros are:
APossible number(s) of positive real zeros: \(\boxed{2, 0}\)
BPossible number(s) of negative real zeros: \(\boxed{2, 0}\)
Examples
Descartes' Rule of Signs is a very practical tool in fields like physics or engineering, where we often work with complex systems described by polynomial equations. For example, if you are designing a control system and the stability of the system depends on the roots of a characteristic polynomial, you can use this rule to quickly estimate how many roots might cause instability (positive roots) without having to perform full, potentially tedious calculations to find every root exactly. This aids in making quick design decisions or identifying potential issues in system behavior before proceeding to more complex numerical analysis.