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 determine the number of possible negative real zeros for a given polynomial, we can utilize Descartes' Rule of Signs.
Step 1: Count the number of sign changes in the coefficients of the polynomial when written in standard form.
For f(x) = 6 + x^4 + 2x^2 - 5x^3 - 12, we have the coefficients:
6 (no sign change),
1 (positive to negative),
2 (negative to positive),
-5 (positive to negative),
-12 (negative).
So, there is only one sign change in the coefficients.
Step 2: Based on step 1, we can conclude either 0 or 2 possible negative real zeros for f(x). However, there cannot be 2 negative real zeros since the degree of the polynomial is 4.
Therefore, the number of possible negative real zeros for f(x) = 6 + x^4 + 2x^2 - 5x^3 - 12 is 0.
Moving on to the second question,
To approximate the real zeros of a polynomial, we can use numerical methods such as graphing or the Newton-Raphson method. However, to determine if there are any real zeros, one approach is to check the discriminant of the polynomial.
For f(x) = 2x^4 - 3x^2 - 2, the discriminant (∆) can be calculated as:
∆ = b^2 - 4ac,
where a, b, and c are the coefficients of the polynomial.
In this case, a = 2, b = 0, and c = -2. Thus,
∆ = (0)^2 - 4(2)(-2) = 16.
The discriminant is positive (∆ > 0). This indicates that there are two distinct irrational or complex roots, but no real roots.
Therefore, the polynomial f(x) = 2x^4 - 3x^2 - 2 has no real zeros.