Answer:
Possible positive real zeros: 3 or 1
Possible negative real zeros: 2 or 0
Step-by-step explanation:
To state the number of possible negative and positive real zeros of the given function, we can use Descartes' Rule of Signs
Descartes' Rule of Signs is a mathematical rule that helps determine the possible number of positive and negative real roots (zeros) of a polynomial by examining the signs of its non-zero consecutive coefficients.
[tex]\hrulefill[/tex]
Positive Real Roots
Examine the non-zero coefficients of the polynomial in descending order, and count the number of sign changes. The number of sign changes gives us the maximum number of possible number of positive real roots.
In the polynomial f(x) = 6x⁵ - 4x⁴ - 63x³ + 42x² + 147x - 98, the sequence of signs is:
[tex]\underbrace{+\; -}_{1}\;\underbrace{-\; +}_{2}\;\underbrace{+\; -}_{3}[/tex]
As there are 3 sign changes, this means that the maximum possible number of positive zeros is 3.
[tex]\hrulefill[/tex]
Negative Real Roots
To determine the maximum possible number of negative roots, first replace each x with -x in the polynomial:
f(-x) = 6(-x)⁵ - 4(-x)⁴ - 63(-x)³ + 42(-x)² + 147(-x) - 98
If (-x) is raised to an odd power, the resulting coefficient is negative, whereas if (-x) is raised to an even power, the resulting coefficient is positive. Therefore, the sequence of signs of the resulting non-zero coefficients in the polynomial f(-x) is:
[tex]-\;\underbrace{-\; +}_{1}\;\underbrace{+\; -}_{2}\;-[/tex]
As there are 2 sign changes, this means that the maximum possible number of negative zeros is 2.
[tex]\hrulefill[/tex]
Some of the roots may be complex, and because of this we have to count down by 2's to find the complete list of the possible number of zeroes.
Therefore:
- Number of possible positive real zeros: 3 or 1
- Number of possible negative real zeros: 2 or 0
