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
How did you arrive at your answers?
To find the number of possible negative real zeros for the polynomial function f(x)=6+x^4+2x^2-5x^3-12, we can use Descartes' rule of signs.
1. First, count the number of sign changes in the coefficients of the terms when we write the polynomial in descending order of powers:
- There is a sign change between 6 and -5x^3, another between -5x^3 and 2x^2, and no more after that.
- So, the number of sign changes is 2.
2. Now, let's find the number of possible positive real zeros. We substitute -x for x in the given polynomial f(x):
- f(-x) = 6+(-x)^4+ 2(-x)^2 -5(-x)^3-12 = 6+x^4+2x^2+5x^3-12
- Counting the sign changes as before, we find that there are no sign changes.
According to Descartes' rule of signs, the number of possible positive real zeros is equal to the number of sign changes or is less than that number by an even integer. Since there are no sign changes in f(-x), there are 0 or 2 positive real zeros.
Since the number of possible negative real zeros is equal to the number of sign changes in f(x) or less than that number by an even integer, we can conclude that there are 0 or 2 negative real zeros. However, since we know that there are 0 real zeros (as stated in the answer), we can conclude that there are 0 possible negative real zeros for f(x)=6+x^4+2x^2-5x^3-12.
Now let's move on to the second question:
To approximate the real zeros of the polynomial function f(x)=2x^4-3x^2-2 to the nearest tenth, we can use numerical methods like the graphing calculator or the Newton-Raphson method. However, before proceeding with those methods, we can first check if there are any rational roots using the rational root theorem.
The rational root theorem states that if a rational number p/q (p and q are integers with no common factors other than 1 and q ≠ 0) is a root of the polynomial equation, then p must be a factor of the constant term (in this case -2), and q must be a factor of the leading coefficient (in this case 2).
In our polynomial function f(x)=2x^4-3x^2-2, the constant term is -2, and the leading coefficient is 2. The factors of -2 are ±1 and ±2, and the factors of 2 are ±1 and ±2.
By testing these possible rational roots (±1, ±2) in the polynomial function, we find that none of them result in f(x)=0. Therefore, there are no rational roots, which means there are no real roots for f(x)=2x^4-3x^2-2.
Hence, the answer is that there are no real roots for f(x)=2x^4-3x^2-2.