Jeremiah has been researching pattern recognition and considering how to clearly communicate his research to potential clients. Jeremiah is interested in the stock market and investing, so he starts watching an online video channel that features tutorials.

Question

One of the videos is about a type of math called business calculus, which analysts use to study trends in the market. He finds many of the symbols and definitions confusing, so he pauses on some parts that he understands and tries to work out a solution. He then plays the video to see if he got it right or to find where he made a mistake.

Part A
During one part of the video, the result of a calculation is a long polynomial expression called P, which needs to be simplified.

P:3(x2−x)+4x−1−5x

Question 1
What are the different types of terms that appear in the polynomial expression?

Question 2
Use the Distributive Property to rewrite and simplify the expression.

Question 3
Use the Commutative Property to rewrite the expression in a different order.

Question 4
Jeremiah rewrites the expression as 3x2−(4x+1)
. Explain clearly why this statement is or is not equivalent to the original expression.

Part B
Jeremiah plays the video again and another polynomial expression, Q, appears. The expression is similar to P, but it has different coefficients.

Question 5
Write an expression Q that is similar to P but with different coefficients.

Question 6
Find the sum of the expressions P+Q
.

Question 7
Find the difference between the expressions P−Q
.

Question 8
Are the expressions P+Q
and P−Q
closed under addition and subtraction, respectively?

Part C
Jeremiah wonders if he can relate the sum and difference of two polynomials using the equation shown.

−(P+Q)=P−Q

Question 9
Are the two expressions shown equivalent to each other? What evidence supports or disproves the statement? Communicate your explanation clearly, remembering to justify your steps by listing the property you used.

Part D
Question 10
Jeremiah finds that the total cost of materials for a rectangular shipping container is defined by the following polynomial division:

35x55x2

What is the simplified form of this expression?

Question 11
Jeremiah determines that the revenue function for a company when it sells x units of a product is defined by the function R(x)=8x4−20x3+40x2
. Part of the calculation of the market price for this product requires the revenue function to be divided by 2x2
. What is the result of this polynomial division?

Part E
Jeremiah reaches the end of the video, where it describes two expressions that need to be multiplied.

X=2x−1
and Y=3x+5

Question 12
What is the product of X⋅Y
?

Question 13
Before you calculated the result, could you have known that the resulting expression is a polynomial?

Answer 1

Let's break down each part of Jeremiah's questions regarding his polynomial research.

Part A

Question 1: What are the different types of terms that appear in the polynomial expression?

In the expression \( P: 3(x^2 - x) + 4x - 1 - 5x \), the types of terms are:

• Constant term: \(-1\)
• Linear terms: \(4x\) and \(-5x\)
• Quadratic term: \(3x^2\)

Question 2: Use the Distributive Property to rewrite and simplify the expression.

To apply the Distributive Property: \[ P = 3(x^2 - x) + 4x - 1 - 5x \] First, distribute \(3\): \[ P = 3x^2 - 3x + 4x - 1 - 5x \] Now combine like terms: \[ P = 3x^2 + (-3x + 4x - 5x) - 1 = 3x^2 - 4x - 1 \]

Question 3: Use the Commutative Property to rewrite the expression in a different order.

Using the Commutative Property, we can rearrange the terms in any order: \[ P = -1 + 3x^2 - 4x \] or \[ P = 3x^2 - 1 - 4x \]

Question 4: Jeremiah rewrites the expression as \( 3x^2 - (4x + 1) \). Explain clearly why this statement is or is not equivalent to the original expression.

The expression \( 3x^2 - (4x + 1) \) is equivalent to the original expression \( 3x^2 - 4x - 1 \) because:

1. When you distribute the negative sign in \( -(4x + 1) \), you get \(-4x - 1\).
2. Therefore, combining gives: \[ 3x^2 - 4x - 1 \] Which is indeed the same as the simplified form of \( P \).

Part B

Question 5: Write an expression \( Q \) that is similar to \( P \) but with different coefficients.

An example of such an expression could be: \[ Q: 2(x^2 - 2x) + 5x + 3 - 4x \]

Question 6: Find the sum of the expressions \( P + Q \).

First, we simplify \( Q \): \[ Q = 2(x^2 - 2x) + 5x + 3 - 4x = 2x^2 - 4x + 5x + 3 - 4x = 2x^2 + 3 - 3x \]

Now, we compute \( P + Q \): \[ P + Q = (3x^2 - 4x - 1) + (2x^2 - 3x + 3) \] Combine like terms: \[ (3x^2 + 2x^2) + (-4x - 3x) + (-1 + 3) = 5x^2 - 7x + 2 \]

Question 7: Find the difference between the expressions \( P - Q \).

Calculating \( P - Q \): \[ P - Q = (3x^2 - 4x - 1) - (2x^2 - 3x + 3) \] Simplifying: \[ 3x^2 - 4x - 1 - 2x^2 + 3x - 3 = (3x^2 - 2x^2) + (-4x + 3x) + (-1 - 3) = x^2 - x - 4 \]

Question 8: Are the expressions \( P + Q \) and \( P - Q \) closed under addition and subtraction, respectively?

Yes, both expressions \( P + Q \) and \( P - Q \) are closed under addition and subtraction, respectively, because the sum and difference of polynomials are still polynomials.

Part C

Question 9: Are the two expressions shown equivalent to each other?

To check \( - (P + Q) = P - Q \): Using a previous result from Questions 6 and 7:

• \( P + Q = 5x^2 - 7x + 2 \)
• Therefore, \( - (P + Q) = - (5x^2 - 7x + 2) = -5x^2 + 7x - 2 \)

Now \( P - Q = x^2 - x - 4 \):

• The two expressions are not equivalent, since: \[
• (P + Q) \neq P - Q \]

Properties used:

• The distributive property to multiply \( -1 \) across \( P + Q \).

Part D

Question 10: What is the simplified form of the polynomial division \( \frac{35}{5x} \)?

Simplifying gives: \[ \frac{35}{5} \cdot \frac{1}{x} = 7 \cdot \frac{1}{x} = \frac{7}{x} \]

Question 11: What is the result of dividing the revenue function \( R(x) = 8x^4 - 20x^3 + 40x^2 \) by \( 2x^2 \)?

Perform the division: \[ \frac{8x^4}{2x^2} - \frac{20x^3}{2x^2} + \frac{40x^2}{2x^2} = 4x^2 - 10x + 20 \]

Part E

Question 12: What is the product of \( X \cdot Y \)?

\[ X = 2x - 1, \quad Y = 3x + 5 \] Using the distributive property (FOIL method): \[ X \cdot Y = (2x - 1)(3x + 5) = (2x)(3x) + (2x)(5) - (1)(3x) - (1)(5) = 6x^2 + 10x - 3x - 5 \] Combining like terms: \[ = 6x^2 + 7x - 5 \]

Question 13: Before you calculated the result, could you have known that the resulting expression is a polynomial?

Yes, the product of two linear polynomials always results in a polynomial, specifically a polynomial of degree equal to the sum of the degrees of the factors. Here, both \( X \) and \( Y \) are linear (degree 1), so their product is a quadratic polynomial (degree 2), confirming the result is a polynomial.

Answer 2

just relay the answeres without the chithcat

Answer 3

Sure! Here are the answers without elaboration:

Part A

1. Constant term: -1; Linear terms: 4x, -5x; Quadratic term: 3x².
2. \( P = 3x^2 - 4x - 1 \).
3. \( P = -1 + 3x^2 - 4x \) or \( P = 3x^2 - 1 - 4x \).
4. It is equivalent; both simplify to \( 3x^2 - 4x - 1 \).

Part B

5. \( Q: 2(x^2 - 2x) + 5x + 3 - 4x \).
6. \( P + Q = 5x^2 - 7x + 2 \).
7. \( P - Q = x^2 - x - 4 \).
8. Yes, \( P + Q \) and \( P - Q \) are closed under addition and subtraction, respectively.

Part C

9. No, \( - (P + Q) \neq P - Q \).

Part D

10. \( \frac{7}{x} \).
11. \( 4x^2 - 10x + 20 \).

Part E

12. \( 6x^2 + 7x - 5 \).
13. Yes, the resulting expression is a polynomial.

Answer 4

edit some of the answers

Answer 5

Sure! Please specify which answers you'd like to edit and how you'd like them changed.

Answer 6

Jeremiah spends several weeks learning and practicing. Then he decides to make his own video on how to rewrite expressions.

Write a short script for Jeremiah to read while he shows an example problem. In the script, include cues for his actions along with the words he will say. Your goal is to communicate the information clearly for an audience by including the following:

descriptions and examples of the different properties
transitions that signal a new step is occurring in the process
active tense and action verbs that illustrate the steps to perform
memorable moments that engage and draw in the audience

Answer 7

Script for Jeremiah’s Video on Rewriting Expressions

[INTRO - Jeremiah stands in front of a whiteboard with a large mathematical expression written on it.]

Jeremiah:
"Hey everyone! I’m Jeremiah, and today we’re diving into the world of rewriting mathematical expressions using key properties. It might feel a bit tricky at first, but I promise you, with practice, you’ll get the hang of it!

[Jeremiah gestures towards the board.]
"Let’s start with the expression \( 3(x^2 - x) + 4x - 1 - 5x \). Our goal is to simplify it step by step. And remember, behind every great simplification is a property at play!

[Step 1 - Applying the Distributive Property]
"First up, we’ll use the Distributive Property. This property allows us to multiply a single term across a group of terms inside parentheses. Watch this!"

[Jeremiah steps closer to the board, pointing at the expression.]
"I'm going to distribute \( 3 \) to both \( x^2 \) and \( -x \). So, we rewrite it as: \[ 3 \cdot x^2 - 3 \cdot x \] Now let’s combine it with the rest of the expression:

\[ 3x^2 - 3x + 4x - 1 - 5x \]

[Step 2 - Combining Like Terms]
"Next, we need to combine like terms. Like terms are terms that have the same variable and exponent. Let’s group them together!”

[Jeremiah uses a different color marker to circle the like terms.]
"Here we have:

• The quadratic term \( 3x^2 \)
• The linear terms: \( -3x + 4x - 5x \)
  And the constant: \( -1 \).

So, we have: \[ 3x^2 + (-3x + 4x - 5x) - 1 \]

[Jeremiah smiles and moves his hands to emphasize the next step.]
"Now, let’s simplify! We combine the linear terms:
\(-3x + 4x - 5x\) gives us \(-4x\).
Thus, the entire expression now looks like this:

\[ 3x^2 - 4x - 1 \]

[MEMORABLE MOMENT - Jeremiah raises his hands excitedly.]
"Ta-da! We did it! That wasn’t so hard, right? This is the beauty of simplifying expressions! You can see how using properties makes things clearer and easier."

[Step 3 - Reordering with the Commutative Property]
"Now, let’s shake things up a little using the Commutative Property. This property allows us to rearrange the terms in any order. Let’s present our expression differently."

[Jeremiah turns the board around, revealing another version of the expression.]
"I'll rewrite it as:
\[ -1 + 3x^2 - 4x \]
See how we can change the order? This still means the same thing!"

[CLOSING - Jeremiah steps back to face the camera, a bright smile on his face.]
"So, that’s how you can rewrite expressions using the Distributive and Commutative Properties! Practice these steps, and you’ll not only simplify expressions but also gain confidence in your math skills. Drop a comment below with your questions or examples you want to tackle together!

Thanks for watching, and happy simplifying!"

[Jeremiah waves goodbye as the video fades out.]