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 real zeros for a polynomial function, we can use the Descartes' rule of signs.
1) For the first question, we have the function f(x) = 6 + x^4 + 2x^2 - 5x^3 - 12. Notice that the function only contains positive coefficients, except for the constant term -12. According to Descartes' rule of signs, the number of possible positive real zeros is either equal to the number of sign changes in f(x) or less than that by an even integer.
Now let's check for sign changes:
From 6 to x^4, we have one sign change (positive to negative).
From x^4 to 2x^2, there is no sign change.
From 2x^2 to -5x^3, there is one sign change (positive to negative).
From -5x^3 to -12, there is no sign change.
So, f(x) has at most one positive real zero. Since it is possible to have zero positive real zeros when there is a constant term, the answer is 0 possible negative real zeros.
2) For the second question, we have the function f(x) = 2x^4 - 3x^2 - 2. To approximate the real zeros, we can use a numerical method like the Newton-Raphson method or graphing the function to observe the x-intercepts.
However, before doing that, we can use the Rational Root Theorem to check for any rational roots. The Rational Root Theorem states that if a polynomial equation with integer coefficients has a rational root p/q, where p and q are relatively prime integers, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
For f(x) = 2x^4 - 3x^2 - 2, the constant term is -2 and the leading coefficient is 2. The factors of the constant term -2 are ±1 and ±2, and the factors of the leading coefficient 2 are ±1 and ±2. Therefore, the possible rational roots are ±1/1, ±2/1, ±1/2, and ±2/2, which simplify to ±1, ±2, ±0.5, and ±1.
By testing these values, we find that none of them are roots of the equation. Therefore, there are no rational roots, which means there are no real roots for f(x) = 2x^4 - 3x^2 - 2.