determine between which consecutive integers the real zeros of each function are located.

f(x)=3x^3 – 10x^2 + 22x – 4 (I got 0,1)
f(x) = x^3 – 12x^2 + 17x – 9 (NO idea)
f(x) = x^3 – 9 (I got 8,9)

Thanks for the help, I really appreciate it. I need the most help with the second one, I couldn't even come up with an answer for that one.

answers were:

0,1
10,11
2,3

To determine between which consecutive integers the real zeros of each function are located, we can use the Intermediate Value Theorem and synthetic division.

1. For the function f(x) = 3x^3 – 10x^2 + 22x – 4:

Step 1: Substitute consecutive integers into the function to find the sign changes.
f(-1) = 3(-1)^3 – 10(-1)^2 + 22(-1) – 4 = -3 + 10 - 22 - 4 = -19
f(0) = 3(0)^3 – 10(0)^2 + 22(0) – 4 = -4
f(1) = 3(1)^3 – 10(1)^2 + 22(1) – 4 = 11

Step 2: Based on the signs of the function values, we can determine between which consecutive integers the real zeros are located. Since f(-1) is negative and f(0) is negative, and f(1) is positive, there must be a zero between -1 and 0, and another zero between 0 and 1. Therefore, the real zeros are located between -1 and 0, and between 0 and 1.

2. For the function f(x) = x^3 – 12x^2 + 17x – 9:

Step 1: Substitute consecutive integers into the function to find the sign changes.
f(-1) = (-1)^3 – 12(-1)^2 + 17(-1) – 9 = -1 + 12 - 17 - 9 = -15
f(0) = (0)^3 – 12(0)^2 + 17(0) – 9 = -9
f(1) = (1)^3 – 12(1)^2 + 17(1) – 9 = -3

Step 2: Based on the signs of the function values, we can determine between which consecutive integers the real zeros are located. Since f(-1) is negative and f(0) is negative, and f(1) is negative, there is no sign change. Therefore, we cannot determine between which consecutive integers the real zeros are located using this method. We may need to use other techniques like the Rational Root Theorem or graphing to find the zeros.

3. For the function f(x) = x^3 – 9:

Step 1: Substitute consecutive integers into the function to find the sign changes.
f(8) = (8)^3 – 9 = 512 – 9 = 503
f(9) = (9)^3 – 9 = 729 – 9 = 720

Step 2: Based on the signs of the function values, we can determine between which consecutive integers the real zeros are located. Since f(8) is positive and f(9) is positive, there is no sign change. Therefore, we cannot determine between which consecutive integers the real zeros are located using this method. In this case, we can see that the function f(x) only has one real zero, which is located between 8 and 9.

To determine between which consecutive integers the real zeros of each function are located, we can use the Intermediate Value Theorem and test the function values at different intervals.

Here's how we can approach each function:

1. f(x) = 3x^3 – 10x^2 + 22x – 4
To find the real zeros of this function, we can start by evaluating f(x) at a few integer values. We want to find intervals where the function changes sign.

Evaluate f(x) at x = -1, 0, and 1:
f(-1) = 3(-1)^3 – 10(-1)^2 + 22(-1) – 4 = -10
f(0) = 3(0)^3 – 10(0)^2 + 22(0) – 4 = -4
f(1) = 3(1)^3 – 10(1)^2 + 22(1) – 4 = 11

From the evaluations, we can see that f(-1) is negative, f(0) is negative, and f(1) is positive. Therefore, there must be a zero between -1 and 0, and another zero between 0 and 1. So, the real zeros are located between the consecutive integers -1 and 0, and between 0 and 1.

2. f(x) = x^3 – 12x^2 + 17x – 9
To find the real zeros of this function, we can follow a similar approach by evaluating f(x) at different integer values.

Evaluate f(x) at x = -1, 0, and 1:
f(-1) = (-1)^3 – 12(-1)^2 + 17(-1) – 9 = -4
f(0) = (0)^3 – 12(0)^2 + 17(0) – 9 = -9
f(1) = (1)^3 – 12(1)^2 + 17(1) – 9 = -3

From the evaluations, we can see that f(-1) is negative, f(0) is negative, and f(1) is negative as well. Since the function does not change sign between -1 and 0, or between 0 and 1, we cannot determine between which consecutive integers the real zeros are located based on this information alone.

To find the real zeros of this function more precisely, we can use methods like factoring, graphing, or numerical approximation techniques.

3. f(x) = x^3 – 9
To find the real zeros of this function, we can again evaluate f(x) at different integer values.

Evaluate f(x) at x = 7, 8, and 9:
f(7) = 7^3 – 9 = 343 – 9 = 334
f(8) = 8^3 – 9 = 512 – 9 = 503
f(9) = 9^3 – 9 = 729 – 9 = 720

From the evaluations, we can see that f(7) is positive, f(8) is positive, and f(9) is positive. Since the function does not change sign between 7 and 8, or between 8 and 9, we cannot determine between which consecutive integers the real zeros are located based on this information alone.

If you need to find the exact numerical values of the real zeros for functions 2 and 3, you can consider factoring or using numerical approximation methods like Newton's method or the bisection method to find the zeros.

Hope this helps! Let me know if you have any further questions.