Hello! I need help with these four math problems. Thanks so much! :)
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
we have
(x^4)/4 − 3x^3 + (23x^2)/2 - 15x
Fractions are a nuisance, so I prefer
x^4 - 12x^3 + 46x^2 - 60x
= x(x^3-12x^2+46x-60)
Cubics are tough, but a little synthetic division shows that we have
x(x-6)(x^2-6x+10)
So, we know we have at most 3 extremes for a 4th-degree polynomial, and that between any two real roots there is at least one extreme.
So, we know there is at least one between 0 and 6.
If all you have is algebra, finding the exact locations can be a chore.
What tools have you learned so far to tackle this kind of problem?
#2
there are 10C7 = 10*9*8 / 1*2*3 ways to answer 7 questions out of 10.
#3
(a-1)^4 = a^4-4a^3+6a^2-4a+1
#4. add all powers of all variables to find the degree.
In this case, the highest is 3xy^3z of degree 1+3+1=5.
Sure! I can help you with these math problems one by one. Here are the step-by-step solutions:
1.) To find the x-values where the extremes occur, we need to find the critical points of the function ƒ(x). Critical points occur where the derivative of the function is equal to zero or undefined.
Step 1: Find the derivative of the function ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x.
ƒ'(x) = (4x^3)/4 - (9x^2)*3 + (23x)*2/2 - 15
Simplifying the derivative, we get:
ƒ'(x) = x^3 - 9x^2 + 23x - 15
Step 2: Set the derivative equal to zero and solve for x:
x^3 - 9x^2 + 23x - 15 = 0
You can use a numerical method or factor the equation to find the solutions. The solutions are x ≈ -0.6801, x ≈ 3.8137, and x ≈ 4.8664.
2.) To calculate the ways you can make a 70% on a 10-question exam where each question is worth 10 points (no partial credit), we need to count the number of ways to attain that total score.
Step 1: Calculate the maximum score on the exam:
Maximum score = 10 (questions) * 10 (points per question) = 100
Step 2: Calculate the minimum score for a 70%:
Minimum score = 70% * Maximum score = 0.7 * 100 = 70
Step 3: Find the number of ways to get a score of 70.
Since each question carries 10 points, the possible scores range from 0 to 100 in increments of 10. To get a total score of 70, there are several possibilities:
- Answer the first 7 questions correctly and leave the remaining 3 questions blank.
- Answer the first 6 questions correctly and leave the last 4 questions blank.
- ...
You can continue listing the number of possible combinations until you find all the ways to achieve a score of 70.
3.) To write the polynomial (a - 1)^4 in standard form and classify it by degree and number of terms, we need to expand it using the binomial theorem.
Step 1: Expand the polynomial using the binomial theorem:
(a - 1)^4 = 1*a^4 + (-4*a^3)*(1) + (6*a^2)*(1^2) + (-4*a)*(1^3) + 1*(1^4)
Simplifying the terms, we get:
(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1
Step 2: Write the polynomial in standard form by arranging the terms in decreasing order of their degree:
(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1
The polynomial has 5 terms and the degree of the polynomial is 4.
4.) To find the degree of the polynomial x^2y − 3xy^3z + 5x + 7y, we need to determine the highest power of the variables.
Step 1: Identify the highest power of each variable in the polynomial:
- The highest power of x is x^2 in the term x^2y.
- The highest power of y is y^3 in the term -3xy^3z.
- There is no z term in the polynomial, so the power of z is 0.
- The term 5x has x raised to the power of 1.
- The term 7y has y raised to the power of 1.
Step 2: Determine the highest power overall:
The highest power of the variables is x^2y^3z^0 = x^2y^3.
Therefore, the degree of the polynomial x^2y − 3xy^3z + 5x + 7y is 2+3 = 5.
I hope this helps! Let me know if you have any further questions.
Of course! I'll be happy to help you with these math problems. Let's go through each problem step by step.
1.) To find the x values where the extremes occur for the given function ƒ(x) = (x^4)/4 − 3x^3 + (23x^2)/2 - 15x, we need to find the critical points. The critical points are the values of x where the derivative of the function is either zero or undefined.
First, we need to find the derivative of ƒ(x) with respect to x. Let's call the derivative ƒ'(x).
ƒ'(x) = (d/dx)(x^4)/4 - (d/dx)(3x^3) + (d/dx)((23x^2)/2) - (d/dx)(15x)
Simplifying this, we get:
ƒ'(x) = x^3 - 9x^2 + 23x/2 - 15
Now, we set ƒ'(x) equal to zero and solve for x:
x^3 - 9x^2 + 23x/2 - 15 = 0
To solve this equation, you can either use numerical methods or factor the equation if possible. Once you find the values of x that make the derivative zero, those are the x values where the extremes occur.
2.) To determine the number of ways to make a 70% on a 10-question exam, where each question is worth 10 points, we need to calculate the number of combinations of correct answers that would result in a score of 70%. Since there are no partial credits given, we need to find the possible combinations that add up to exactly 70 points.
To solve this problem, we can use the concept of a "stars and bars" or "balls and urns" problem. In this case, consider each question as a "star," and we need to allocate a total of 70 points among these 10 questions. The "bars" or "urns" represent the divisions between each question.
To find the number of ways, we can use the formula for combinations. The formula is (n + k - 1) choose (k - 1), where n is the total number of points (70 in this case) and k is the number of questions (10 in this case).
So, the number of ways to make a 70% on the exam is (70 + 10 - 1) choose (10 - 1) or (79 choose 9). You can calculate this using a calculator or a combination calculator.
3.) The given polynomial is (a - 1)^4. To write it in standard form, we need to expand and simplify the expression.
Using the binomial theorem or simply multiplying out the terms, we have:
(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1
Now, let's classify it by degree and number of terms. The degree of the polynomial is the highest exponent of the variable, which in this case is 4. The number of terms is the total number of different terms in the polynomial, which is 5 in this case.
So, the polynomial (a - 1)^4 is a 5-term polynomial of degree 4.
4.) The given polynomial is x^2y − 3xy^3z + 5x + 7y.
To find the degree of the polynomial, we need to determine the highest total power of any single term in the polynomial.
In this case, the highest total power is x^2y, which has a degree of 2 + 1 = 3 (since the degree is the sum of the exponents of each variable in the term).
Therefore, the degree of the polynomial x^2y − 3xy^3z + 5x + 7y is 3.
I hope this helps you solve your math problems! Let me know if you have any further questions.