Find the number of possible negative real zeros for f(x)=6+x^4+2x^2-5x^3-12.
Answer: 0
2) Approximate the real zeros of f(x)=2x^4-3x^2-2 to the nearest tenth.
Answer: no real roots
To find the number of possible negative real zeros for the function f(x)=6+x^4+2x^2-5x^3-12, we can use the Rule of Signs.
1. First, we need to determine the number of sign changes in the coefficients when we write the polynomial in descending order.
The given polynomial can be rearranged as: f(x)=x^4-5x^3+2x^2+6-12.
Now, let's check the sign changes:
a) From the term x^4 to -5x^3, there is one sign change.
b) From -5x^3 to +2x^2, there is no sign change.
c) From +2x^2 to +6, there is no sign change.
d) From +6 to -12, there is one sign change.
Thus, we have 2 sign changes in the coefficients.
2. Now let's consider f(-x). By replacing x with -x in the polynomial, we get f(-x)=(-x)^4-5(-x)^3+2(-x)^2+6-12.
Simplifying this gives: f(-x)=x^4+5x^3+2x^2-6x-6.
Again, we count the number of sign changes in the coefficients:
a) From the term x^4 to +5x^3, there is one sign change.
b) From +5x^3 to +2x^2, there is no sign change.
c) From +2x^2 to -6x, there is one sign change.
d) From -6x to -6, there is no sign change.
Thus, we have 2 sign changes in the coefficients of f(-x).
3. According to the Rule of Signs, the number of possible negative real zeros is equal to the number of sign changes or less by an even integer. In this case, we have 2 sign changes, so the number of possible negative real zeros is either 2 or less than 2 by an even integer.
However, since the given polynomial has no negative real zeros, we conclude that the number of possible negative real zeros for f(x)=6+x^4+2x^2-5x^3-12 is 0.
Now, let's move on to the second question.
To approximate the real zeros of the function f(x)=2x^4-3x^2-2 to the nearest tenth, we can use numerical methods like the Newton-Raphson method or the bisection method. However, we can also examine the polynomial directly to determine if it has any real zeros.
By analyzing the coefficients of the polynomial, we can see that there are no sign changes in the coefficients. Since the coefficient of the highest power term is positive, and there are no sign changes, we can conclude that there are no real zeros for this polynomial.
Therefore, the answer is that f(x)=2x^4-3x^2-2 has no real roots.