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During one part of the video, the result of a calculation is a long polynomial expression called P

that needs to be simplified.

P:3(x2−x)+4x−5x−1

Question 1

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

Question 2

Use the Distributive Property and Associative Property to rewrite and simplify the expression.

Question 3

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

Question 4

Jeremiah writes 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 then another polynomial expression, Q
, appears. The expression is similar to P
, but 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 of the expressions P−Q
.

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

−(P+Q)=P−Q

Question 8

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
Jeremiah notices there are several examples of polynomials being divided in this presentation. In particular, he notices that second-degree polynomials are sometimes divided by first-degree polynomials, and others by second-degree polynomials.

Question 9

At one point in the presentation, the second-degree polynomial 4x2+20x−75
is divided by the first-degree polynomial 2x−5
. What is the quotient? Show your work.

Question 10

At another point in the presentation, the second-degree polynomial 9x2+18x+6
is divided by the second-degree polynomial 3x2+6x+2
. What is the quotient? Show your work.

1 answer

Let's break down the questions step by step.

Part A

Question 1

The polynomial expression \( P = 3(x^2 - x) + 4x - 5x - 1 \) consists of the following types of terms:

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

So, \( P \) consists of one constant term, one quadratic term, and two linear terms.

Question 2

Use the Distributive Property and Associative Property to rewrite and simplify the expression:

First, use the Distributive Property on the first part: \[ 3(x^2 - x) = 3x^2 - 3x \] Now replace this in the original expression: \[ P = (3x^2 - 3x) + 4x - 5x - 1 \] Now combine like terms (use the Associative Property to group them together): \[ P = 3x^2 + (-3x + 4x - 5x) - 1 \] Now simplify the linear terms: \[ -3x + 4x - 5x = -4x \] Thus, \[ P = 3x^2 - 4x - 1 \]

Question 3

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

We can change the order of the terms in \( P \): \[ P = -1 + 3x^2 - 4x \]

Question 4

Assessment of the expression \( 3x^2 - (4x + 1) \):

Let's simplify \( 3x^2 - (4x + 1) \): \[ 3x^2 - (4x + 1) = 3x^2 - 4x - 1 \]

Since this results in the same expression as \( P = 3x^2 - 4x - 1 \), we can conclude that:

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

Part B

Question 5

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

For example: \[ Q = 5(x^2 - 2x) + 2x - 3x - 4 \]

Question 6

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

We know \( P = 3x^2 - 4x - 1 \) and we find \( Q \): \[ Q = 5(x^2 - 2x) + 2x - 3x - 4 = 5x^2 - 10x + 2x - 3x - 4 = 5x^2 - 11x - 4 \] Now sum them up: \[ P + Q = (3x^2 - 4x - 1) + (5x^2 - 11x - 4) \] Combine like terms: \[ (3x^2 + 5x^2) + (-4x - 11x) + (-1 - 4) = 8x^2 - 15x - 5 \]

Question 7

Find the difference of the expressions \( P - Q \):

Continuing with \( P \) and \( Q \): \[ P - Q = (3x^2 - 4x - 1) - (5x^2 - 11x - 4) \] Distributing the negative: \[ = 3x^2 - 4x - 1 - 5x^2 + 11x + 4 \] Combine like terms: \[ (3x^2 - 5x^2) + (-4x + 11x) + (-1 + 4) = -2x^2 + 7x + 3 \]

Part C

Question 8

Check whether \(- (P + Q) = P - Q\):

Calculate \( -(P + Q) \): \[ P + Q = 8x^2 - 15x - 5 \implies - (P + Q) = -8x^2 + 15x + 5 \] Now calculate \( P - Q \): \[ P - Q = -2x^2 + 7x + 3 \] Clearly, \(- (P + Q) \neq P - Q\).

Evidence:

  • The left side is \(-8x^2 + 15x + 5\) and the right side is \(-2x^2 + 7x + 3\). Since their coefficients are different, the two expressions are not equivalent.

Part D

Question 9

Divide the polynomial \( 4x^2 + 20x - 75 \) by \( 2x - 5 \).

Using polynomial long division:

  1. Divide the leading term: \( \frac{4x^2}{2x} = 2x \)
  2. Multiply: \( 2x(2x - 5) = 4x^2 - 10x \)
  3. Subtract: \[ (4x^2 + 20x - 75) - (4x^2 - 10x) = 30x - 75 \]
  4. Divide the leading term of the remainder: \( \frac{30x}{2x} = 15 \)
  5. Multiply: \( 15(2x - 5) = 30x - 75 \)
  6. Subtract: \( 30x - 75 - (30x - 75) = 0 \)

So, the quotient is \( 2x + 15 \).

Question 10

Divide the polynomial \( 9x^2 + 18x + 6 \) by \( 3x^2 + 6x + 2 \).

Using polynomial long division:

  1. Divide leading terms: \( \frac{9x^2}{3x^2} = 3 \)
  2. Multiply: \( 3(3x^2 + 6x + 2) = 9x^2 + 18x + 6 \)
  3. Subtract: \[ (9x^2 + 18x + 6) - (9x^2 + 18x + 6) = 0 \]

Thus, \( 9x^2 + 18x + 6 \div (3x^2 + 6x + 2) = 3 \) with no remainder.

Summary

  1. Different types of terms in \( P \).
  2. \( P \) simplified to \( 3x^2 - 4x - 1 \).
  3. \( P \) rearranged: \(-1 + 3x^2 - 4x\).
  4. \( 3x^2 - (4x + 1) \) is equivalent to original expression.
  5. Example \( Q \).
  6. Sum \( P + Q = 8x^2 - 15x - 5 \).
  7. Difference \( P - Q = -2x^2 + 7x + 3 \).
  8. \( -(P + Q) \neq P - Q \).
  9. Quotient of \( \frac{4x^2 + 20x - 75}{2x - 5} = 2x + 15 \).
  10. Quotient of \( \frac{9x^2 + 18x + 6}{3x^2 + 6x + 2} = 3 \).