f(x)=2x^4+5x^2-6x+9
Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros.
positive: two sign changes, max two real roots.
negative: zero sign changes, zero real roots
min number of non-real roots:
4-(2-0)=2
one flip in f(x) from 5 x^2 to - 6 x
second from -6 x to +9
no flips in f(-x) = 2x^4 + 5x^2 + 6x + 9
so two or zero positive real roots
check with
http://www.emathhelp.net/calculators/algebra-1/descartes-rule-of-signs-calculator/
To apply Descartes' Rule of Signs to the function f(x) = 2x^4 + 5x^2 - 6x + 9, we will count the number of sign changes in the coefficients.
1. Count the sign changes in the original function:
The original function f(x) = 2x^4 + 5x^2 - 6x + 9 has 2 sign changes.
2. Count the sign changes in the function obtained by replacing x with -x:
Replacing x with -x, we get f(-x) = 2(-x)^4 + 5(-x)^2 - 6(-x) + 9
Simplifying, f(-x) = 2x^4 + 5x^2 + 6x + 9
The function f(-x) has 0 sign changes.
Based on Descartes' Rule of Signs:
- The number of positive real zeros of the function f(x) is either 2 or 0 (the number of sign changes).
- The number of negative real zeros of the function f(x) is either 2 or a multiple of 2 (the number of sign changes or subtracting a multiple of 2).
Therefore, the possible numbers of positive real zeros are 2 or 0, and the possible numbers of negative real zeros are 2, 0, or any even number greater than 2.
To apply Descartes' Rule of Signs, we need to identify the changes in sign (+ to - or - to +) within the coefficients of the polynomial function.
Step 1: Count the sign changes in the polynomial.
Start by listing the coefficients of the terms in descending order:
a = +2, b = +5, c = -6, d = +9
Count the sign changes:
From a (+2) to b (+5) there is no sign change.
From b (+5) to c (-6) there is a sign change.
From c (-6) to d (+9) there is a sign change.
Step 2: Determine the maximum possible number of positive real zeros.
The number of sign changes in the polynomial function f(x) represents the maximum possible number of positive real zeros.
In this case, there is only one sign change, so there can be at most one positive real zero.
Step 3: Apply the rule for negative real zeros.
To determine the maximum possible number of negative real zeros, we substitute -x for x in the polynomial function f(-x).
The polynomial becomes: f(-x) = 2(-x)^4 + 5(-x)^2 - 6(-x) + 9 = 2x^4 + 5x^2 + 6x + 9.
Repeat step 1 for the modified polynomial:
a = +2, b = +5, c = +6, d = +9
Count the sign changes:
From a (+2) to b (+5) there is no sign change.
From b (+5) to c (+6) there is no sign change.
From c (+6) to d (+9) there is no sign change.
Step 4: Determine the maximum possible number of negative real zeros.
Since there are no sign changes in the modified polynomial function, there can be at most zero negative real zeros.
In summary, according to Descartes' Rule of Signs:
- The maximum number of positive real zeros is 1.
- The maximum number of negative real zeros is 0.
Note that Descartes' Rule of Signs only provides an upper bound on the number of positive and negative real zeros. The actual number of real zeros may be fewer than the maximum. To find the exact number and values of real zeros, additional methods like the Rational Root Theorem, synthetic division, or graphing may be necessary.