1.) ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x. Find the x values where the extremes occur.


2.) On a 10 question exam each question is worth 10 points (no partial credit). How ways can you make a 70%?

3.) Write the polynomial in standard form. Then classify it by degree and number of terms. (a - 1)^4

4.) Find the degree of the polynomial below. x^2y − 3xy^3z + 5x + 7y

1.) To find the x values where the extremes occur in the function ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x, we need to find the critical points.

First, let's find the derivative of the function:
ƒ'(x) = (d/dx)((x^4)/4) - (d/dx)(3x^3) + (d/dx)((23x^2)/2) - (d/dx)(15x)
ƒ'(x) = (4x^3)/4 - (9x^2) + (23x)/2 - 15

Next, we set the derivative equal to zero and solve for x to find the critical points:
(4x^3)/4 - (9x^2) + (23x)/2 - 15 = 0

Simplifying the equation, we get:
x^3 - 9x^2 + (23x)/2 - 15 = 0

Unfortunately, solving this equation algebraically can be challenging. We can use numerical methods or calculators to approximate the critical points. Some common numerical methods include graphing the function and observing where the slope changes from positive to negative or using the Newton-Raphson method.

2.) To determine the number of ways you can make a 70% on a 10-question exam, let's first calculate the maximum possible score. Each question is worth 10 points, so the maximum total score is 10 x 10 = 100.

To achieve a 70% on the exam, you would need to score 70% of the maximum total score, which is 0.70 x 100 = 70 points.

Now, we need to find the different combinations of question scores that sum up to 70. Since there is no partial credit and each question is worth 10 points, there are limited options: you can either answer 7 questions correctly and 3 questions incorrectly or answer 8 questions correctly and 2 questions incorrectly.

Therefore, there are 2 ways to make a 70% on the exam.

3.) The polynomial (a - 1)^4 can be expanded using the binomial expansion or Pascal's Triangle. Let's simplify it and then classify it by degree and number of terms.

(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1

The simplified polynomial in standard form is a^4 - 4a^3 + 6a^2 - 4a + 1.

This polynomial has 5 terms and a degree of 4, making it a quartic polynomial.

4.) To find the degree of the polynomial x^2y - 3xy^3z + 5x + 7y, we need to determine the highest exponent of any variable.

The highest exponent is 3, which appears in the term -3xy^3z. Therefore, the degree of the polynomial is 3.

1.) To find the x values where the extremes occur for the function ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x, we need to find the critical points. Critical points occur where the derivative of the function is equal to zero or undefined.

First, let's find the derivative of the function:
ƒ'(x) = (d/dx)((x^4)/4 − 3x^3 + (23x^2)/2 - 15x)

Taking the derivative term by term:
ƒ'(x) = (4x^3)/4 - (9x^2) + (23x)/2 - 15

Next, set the derivative equal to zero and solve for x:
(4x^3)/4 - (9x^2) + (23x)/2 - 15 = 0

Simplifying the equation:
x^3 - (9x^2)/4 + (23x)/8 - 15/4 = 0

Unfortunately, this equation does not have a simple solution. To find the critical points, you'll need to solve this equation numerically using methods like numerical approximation or a graphing calculator.

2.) To find the number of ways to make a 70% on a 10-question exam where each question is worth 10 points, we can use a combination of counting methods.

To get a score of 70%, you need to earn 70 out of 100 points. In this scenario, each question is worth 10 points, so you need to answer 7 questions correctly.

Using combinations, we can calculate the number of ways to choose 7 questions out of 10:
10C7 = 10! / (7! * (10-7)!) = 10! / (7! * 3!)

Solving this equation will give us the total number of ways to make a 70% on the exam.

3.) The polynomial (a - 1)^4 can be written in standard form by expanding it:

(a - 1)^4 = (a - 1)(a - 1)(a - 1)(a - 1)

Expanding this polynomial will give us:

(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1

The polynomial (a - 1)^4 has a degree of 4 and 5 terms.

4.) The polynomial x^2y − 3xy^3z + 5x + 7y can be classified by its degree and number of terms.

Degree: To find the degree of the polynomial, we determine the highest exponent of variable(s) in the polynomial.

In this polynomial, we can see that the highest exponent for any variable is 3 in the term -3xy^3z. Therefore, the degree of the polynomial is 3.

Number of terms: To find the number of terms in a polynomial, count the number of distinct expressions separated by addition or subtraction.

In the given polynomial, there are four distinct expressions separated by addition and subtraction, which are x^2y, -3xy^3z, 5x, and 7y. Therefore, the polynomial has 4 terms.