How many zeros does the function f(x)=-3x^4+2x^3+7x-5 Have?

Question

Can any Expert that answer this question add all steps please.

Answer 1

Without graphing the function, the degree will determine how many zeros the equation will have. Since the degree (the biggest exponent) of this given function is 4, then it has 4 roots. The next question is what kind of roots does the function have? If the graph crosses the x-axis 4 times, then there will be 4 real roots. If it only crosses 2 times, then it has 2 real roots and 2 complex roots. If it doesn’t cross the x-axis at all, then it has 0 real roots and 4 complex roots.

Answer 2

Mike, as you see from the variety of correct answers from Tabitha, David, Peter, Obinna, and Dianne, the wording of the question is slightly vague.

1) number of real roots

2) number of roots, real and complex combined

3) number of roots and their multiplicities

could all be different.

The tutors all agree about the facts, they differ only in their interpretation of the words “zeroes” and “roots”.

For this equation,

Number of real roots/zeroes = 2

Number of roots/zeroes, including complex = 4

With some more analysis, one can see that all roots have multiplicity 1, so there really are 2 complex zeroes/roots.

Answer 3

Answer 4

The question was how many zeros does the quartic polynomial function f(x) = -3x^4+2x^3+7x-5 

have. That is a different question from the maximum number of zeros that any quartic function might have.

This function has 2 zeros. Evaluate at 0 and 1 you get -5 and a positive value. You get a negative and a positive f(x) value. There is a root between 0 and 1!

Descartes rule of signs tells us that there are either 0 or 2 positive real roots, so there must be 2 roots.

Using Descartes rule of signs again for negative values, it is clear that there are 0 negative real roots.

Thus, the total is 2 real roots.

The question asked about zeroes. I assumed that that meant values of x for which the function crossed the real x axis.

Answer 5

The zero of a function can also be called the x-intercept of the function. So you can easily find the zeros of any function by graphing it. Using a graphing calculator or www.desmos.com, graph the function above to see that it crosses the x-axis exactly twice... so 2 zeros.

Answer 6

Highest number of zeros for a function is the highest exponent. In your example that would be 4