Determine which consecutive integers the real zeros of the function are located.

f(x) = 4x^4 - 16x^3 - 25x^2 + 196x -146

Is there an easier way to this besides trial and error synthetic division?

Also, how do I approximate the real zeros?

If the zeroes are the result of whole numbers(X = 1,2,3,and etc),I eliminate

the trial and error portion by using
EXCEL spread sheets to find the zeroes; but in order to show your work, you will
probably have to use synthetic division.

If the zeroes are the result of whole numbers(X = 1,2,3,and etc),I eliminate

the trial and error portion by using
EXCEL spread sheets to find the zeroes; but in order to show your work, you will probably have to use synthetic division.

You should double check your Eq, because the 2 zeroes that I found
were not the result of whole X values:
(-3.321 , 0.0) , (0.891 , 0.0).
I inputed values of X ranging from -10 to +10 and only found the two zeroes shown above. They would be almost
impossible to find by trial and error.

Good luck!

To determine the consecutive integers where the real zeros of the function are located, there are a few methods that can simplify the process.

One method is to use the Rational Root Theorem. According to the theorem, if a rational number p/q is a zero of the polynomial, it must satisfy the following criteria:
- p is a factor of the constant term (in this case, -146)
- q is a factor of the leading coefficient (in this case, 4)

So, the rational roots that we need to check will have the form of p/q, with p being a factor of -146 and q being a factor of 4.

Another method is to use a graphing calculator or software to plot the function and visually identify the x-intercepts or zeros.

To approximate the real zeros of the function, you can use numerical methods like the Newton-Raphson method or the bisection method. These methods involve iteratively refining an initial guess to get closer and closer approximations to the zeros. However, implementing these numerical methods requires knowledge of calculus.

If you don't want to use trial and error synthetic division or calculus-based methods, the Rational Root Theorem provides a systematic approach to identify potential rational zeros. From there, you can test these potential zeros using synthetic division or other methods to find the real zeros.

To determine the consecutive integers where the real zeros of the function are located, you can use a combination of factoring and synthetic division. However, to make the process easier and more efficient, you can use the Rational Root Theorem.

The Rational Root Theorem states that if a polynomial has a rational zero 𝑃/𝑄, where 𝑃 is a factor of the constant term and 𝑄 is a factor of the leading coefficient, then the real zeros of the polynomial are among the possible values obtained by dividing all the factors of the constant term by the factors of the leading coefficient.

In the given function, f(x) = 4x^4 - 16x^3 - 25x^2 + 196x -146, the constant term is -146, and the leading coefficient is 4.

To find the possible rational zeros, you need to consider all the factors of -146 and divide them by the factors of 4. The factors of -146 are ±1, ±2, ±73, ±146, and the factors of 4 are ±1, ±2, and ±4. Therefore, the possible rational zeros are:

±1/1, ±2/1, ±73/1, ±146/1, ±1/2, ±2/2, ±73/2, ±146/2, ±1/4, ±2/4, ±73/4, ±146/4

Simplifying these fractions, the possible rational zeros are:

±1, ±2, ±73, ±146, ±1/2, ±73/2, ±1/4, ±73/4

Now, to approximate the real zeros of the function, you can use numerical methods such as Newton's method or the Bisection method. These methods involve repeatedly applying calculations to approximate the zeros of the function.

Newton's method involves an initial guess for the zero and then refining the approximation through several iterations. The Bisection method involves an interval where the zero is known to exist and successively narrowing down the interval.

Both of these methods require additional calculations that may be beyond the scope of this response. However, there are various online calculators and software that can help you approximate the real zeros of a function. You can input the function and let the software do the calculations for you.