Use the graph to shorten the list of possible rational zeros of the function. Then find all real zeros of the function.

1.) f(x)= 4x^3-8x^2-15x+9

4.) f(x)= 2x^3-5x^2-4x+10

who knows how to do this?

checking the graphs at wolframalpha.com,

http://www.wolframalpha.com/input/?i=4x^3-8x^2-15x%2B9

http://www.wolframalpha.com/input/?i=2x^3-5x^2-4x%2B10

it is clear that
#1 has a root at x = 3
#2 has a root at x = 5/2

So, divide through and you can use the quadratic formula to find the other roots.

the answer is wherever the graph hits the x axis.

1. -3/4, 1/2, 3
2. -3/2, 3/2, 5/2

To find the possible rational zeros of a polynomial function, we can use the Rational Root Theorem. According to this theorem, if the polynomial has a rational zero p/q, where p is a factor of the constant term (in this case, the last term) and q is a factor of the leading coefficient (in this case, the coefficient of the highest power term), then p can be any factor of the constant term, and q can be any factor of the leading coefficient.

For the first function, f(x) = 4x^3 - 8x^2 - 15x + 9:

1. Determine the factors of the constant term (9): ±1, ±3, ±9.
2. Determine the factors of the leading coefficient (4): ±1, ±2, ±4.
3. Generate the possible rational zeros by taking all combinations of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient: ±1/1, ±1/2, ±1/4, ±3/1, ±3/2, ±3/4, ±9/1, ±9/2, ±9/4.
4. Simplify the rational zeros: ±1, ±1/2, ±1/4, ±3, ±3/2, ±3/4, ±9, ±9/2, ±9/4.

Now we can find the real zeros by testing these possible rational zeros using synthetic division or by applying the Factor Theorem.

For the second function, f(x) = 2x^3 - 5x^2 - 4x + 10:

1. Determine the factors of the constant term (10): ±1, ±2, ±5, ±10.
2. Determine the factors of the leading coefficient (2): ±1, ±2.
3. Generate the possible rational zeros by taking all combinations of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient: ±1/1, ±1/2, ±2/1, ±2/2, ±5/1, ±5/2, ±10/1, ±10/2.
4. Simplify the rational zeros: ±1, ±1/2, ±2, ±5, ±5/2, ±10, ±5.

Again, test these possible rational zeros to find the real zeros using synthetic division or the Factor Theorem.

Keep in mind that finding the real zeros may require multiple trials, and there might be additional irrational or complex zeros that cannot be determined by the Rational Root Theorem alone.