Answer :
Sure! Let's determine how many negative real zeros the polynomial [tex]\( f(x) = 3x^5 - 10x^4 + 12x^3 - 15x^2 + 25x - 35 \)[/tex] has using Descartes' rule of signs.
To apply Descartes' rule of signs to find the number of negative real zeros, we need to look at the polynomial [tex]\( f(-x) \)[/tex]. This is done by substituting [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the original polynomial and then checking for changes in sign of the coefficients.
Step-by-step Solution:
1. Substitute [tex]\(-x\)[/tex] into the polynomial:
[tex]\[
f(-x) = 3(-x)^5 - 10(-x)^4 + 12(-x)^3 - 15(-x)^2 + 25(-x) - 35
\][/tex]
2. Simplify the terms:
- [tex]\(3(-x)^5 = -3x^5\)[/tex]
- [tex]\(-10(-x)^4 = -10x^4\)[/tex]
- [tex]\(12(-x)^3 = -12x^3\)[/tex]
- [tex]\(-15(-x)^2 = -15x^2\)[/tex]
- [tex]\(25(-x) = -25x\)[/tex]
- The constant term remains [tex]\( -35 \)[/tex]
Combining all these, we get:
[tex]\[
f(-x) = -3x^5 - 10x^4 - 12x^3 - 15x^2 - 25x - 35
\][/tex]
3. Identify the sign changes:
Now, observe the sequence of coefficients in [tex]\( f(-x) \)[/tex]: [tex]\(-3, -10, -12, -15, -25, -35\)[/tex].
Since all the coefficients are negative, there are no sign changes in this sequence.
4. Apply Descartes' Rule of Signs:
Descartes' Rule of Signs states that the number of negative real zeros of the polynomial is equal to the number of sign changes in [tex]\( f(-x) \)[/tex]. In this case, since there are no sign changes, the number of negative real zeros is zero.
Therefore, the polynomial [tex]\( f(x) = 3x^5 - 10x^4 + 12x^3 - 15x^2 + 25x - 35 \)[/tex] has no negative real zeros.
To apply Descartes' rule of signs to find the number of negative real zeros, we need to look at the polynomial [tex]\( f(-x) \)[/tex]. This is done by substituting [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the original polynomial and then checking for changes in sign of the coefficients.
Step-by-step Solution:
1. Substitute [tex]\(-x\)[/tex] into the polynomial:
[tex]\[
f(-x) = 3(-x)^5 - 10(-x)^4 + 12(-x)^3 - 15(-x)^2 + 25(-x) - 35
\][/tex]
2. Simplify the terms:
- [tex]\(3(-x)^5 = -3x^5\)[/tex]
- [tex]\(-10(-x)^4 = -10x^4\)[/tex]
- [tex]\(12(-x)^3 = -12x^3\)[/tex]
- [tex]\(-15(-x)^2 = -15x^2\)[/tex]
- [tex]\(25(-x) = -25x\)[/tex]
- The constant term remains [tex]\( -35 \)[/tex]
Combining all these, we get:
[tex]\[
f(-x) = -3x^5 - 10x^4 - 12x^3 - 15x^2 - 25x - 35
\][/tex]
3. Identify the sign changes:
Now, observe the sequence of coefficients in [tex]\( f(-x) \)[/tex]: [tex]\(-3, -10, -12, -15, -25, -35\)[/tex].
Since all the coefficients are negative, there are no sign changes in this sequence.
4. Apply Descartes' Rule of Signs:
Descartes' Rule of Signs states that the number of negative real zeros of the polynomial is equal to the number of sign changes in [tex]\( f(-x) \)[/tex]. In this case, since there are no sign changes, the number of negative real zeros is zero.
Therefore, the polynomial [tex]\( f(x) = 3x^5 - 10x^4 + 12x^3 - 15x^2 + 25x - 35 \)[/tex] has no negative real zeros.
