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Multiply the following polynomials and write the resulting polynomials in descending order.

(2x−7)(3x2+7x−8)
(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The value of the coefficient of the x3
is Response area

The value of the coefficient of the x2
is Response area
The value of the coefficient of the x
is Response area
The value of the coefficient of the constant is Response area

11 answers

To multiply the polynomials \( (2x - 7) \) and \( (3x^2 + 7x - 8) \), we can use the distributive property (also known as the FOIL method for binomials). Let's carry out the multiplication step by step:

\[ (2x - 7)(3x^2 + 7x - 8) \]

Distributing \( 2x \) to each term in \( (3x^2 + 7x - 8) \): \[ 2x \cdot 3x^2 = 6x^3 \] \[ 2x \cdot 7x = 14x^2 \] \[ 2x \cdot (-8) = -16x \]

Distributing \( -7 \) to each term in \( (3x^2 + 7x - 8) \): \[ -7 \cdot 3x^2 = -21x^2 \] \[ -7 \cdot 7x = -49x \] \[ -7 \cdot (-8) = 56 \]

Now combine all the results: \[ 6x^3 + 14x^2 - 16x - 21x^2 - 49x + 56 \]

Next, we consolidate the like terms:

  • The \( x^2 \) terms: \( 14x^2 - 21x^2 = -7x^2 \)
  • The \( x \) terms: \( -16x - 49x = -65x \)

So we can write the result in descending order: \[ 6x^3 - 7x^2 - 65x + 56 \]

Now that we have the polynomial, we can answer the questions about the coefficients:

  1. The value of the coefficient of the \( x^3 \) term is 6.
  2. The value of the coefficient of the \( x^2 \) term is -7.
  3. The value of the coefficient of the \( x \) term is -65.
  4. The value of the coefficient of the constant term is 56.

Here are the responses summarized:

  • The value of the coefficient of the \( x^3 \) is 6.
  • The value of the coefficient of the \( x^2 \) is -7.
  • The value of the coefficient of the \( x \) is -65.
  • The value of the coefficient of the constant is 56.
Eric multiplied (6x2−7x+4.7)(x2−2.8)
using an area model.

Complete the blank spaces in the area model and determine the product in standard form.

(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
6x2
−7x
4.7
x2
Response area −7x3
Response area
−2.8
Response area

To find the product of the two polynomials \( (6x^2 - 7x + 4.7)(x^2 - 2.8) \) using an area model, we will create a grid based on the terms of each polynomial and fill in the resulting products of each combination of terms.

Setting Up the Area Model:

We can break down the two polynomials as follows:

  1. First Polynomial: \( 6x^2, -7x, 4.7 \)
  2. Second Polynomial: \( x^2, -2.8 \)

This will create a 3x2 area model.

First Row (Multiplying by \( x^2 \)):

  • \( 6x^2 \cdot x^2 = 6x^{4} \)
  • \( -7x \cdot x^2 = -7x^{3} \)
  • \( 4.7 \cdot x^2 = 4.7x^{2} \)

Second Row (Multiplying by \( -2.8 \)):

  • \( 6x^2 \cdot -2.8 = -16.8x^{2} \)
  • \( -7x \cdot -2.8 = 19.6x \)
  • \( 4.7 \cdot -2.8 = -13.16 \)

Filling in the Blank Spaces:

Now we can fill in the blanks for the area model:

  • First Row: \( 6x^2, -7x, 4.7 \) multiplied by \( x^2 \):

    • Area for \( x^2 \):
      • \( 6x^4 \)
      • \(-7x^3\)
      • \( 4.7x^2 \)

  • Second Row: \( 6x^2, -7x, 4.7 \) multiplied by \( -2.8 \):

    • Area for \( -2.8 \):
      • \(-16.8x^2\)
      • \(19.6x\)
      • \(-13.16\)

Final Result:

Now we can sum the individual products to determine the final polynomial:

  1. Combine all the terms:

    \[ 6x^{4} + (-7x^{3}) + (4.7x^{2} - 16.8x^{2}) + 19.6x - 13.16 \]

  2. Simplify the like terms:

    • For \( x^{2} \): \( 4.7 - 16.8 = -12.1x^2 \)

The final product in standard form is: \[ 6x^{4} - 7x^{3} - 12.1x^{2} + 19.6x - 13.16 \]

Now to fill out the response areas:

  • The area for \( x^2 \): \(-7x^3\)
  • The area for \(-2.8\): \(-16.8x^{2}\)

Thus, for your question:

  1. Area for \(-7x^3\): \(-7x^3\)
  2. Area for \(-16.8x^2\): \(-16.8x^{2}\)

The final polynomial product is: \[ 6x^4 - 7x^3 - 12.1x^2 + 19.6x - 13.16 \]

6x^2 x x^2

To calculate the product of \(6x^2\) and \(x^2\):

\[ 6x^2 \cdot x^2 = 6 \cdot (x^2 \cdot x^2) = 6 \cdot x^{2+2} = 6x^4 \]

So, \(6x^2 \cdot x^2 = 6x^4\).

If you need any further operations or have different calculations in mind, please let me know!

-2.8 x 6x^2

To calculate the product of \(-2.8\) and \(6x^2\):

\[ -2.8 \cdot 6x^2 = (-2.8 \cdot 6) \cdot x^2 = -16.8x^2 \]

So, the product of \(-2.8\) and \(6x^2\) is \(-16.8x^2\). If you have any more calculations or questions, feel free to ask!

-7x(-2.8)

To calculate the product of \(-7x\) and \(-2.8\):

\[ -7x \cdot (-2.8) = 7 \cdot 2.8 \cdot x \]

Calculating \(7 \cdot 2.8\):

\[ 7 \cdot 2.8 = 19.6 \]

Thus, the product is:

\[ 7 \cdot 2.8 \cdot x = 19.6x \]

So, the result of \(-7x \cdot -2.8\) is \(19.6x\). If you have any more questions or calculations, feel free to ask!

Two students in Ms. Chau's class were asked to write an equivalent expression for the expression (a−6)(a−4)
. Identify the student with the correct work and where the mistakes were in the other student's work.

(1 point)
Responses

Gabrielle is correct; Brittany multiplied the constants −6⋅−4
and got +24
, and she should have gotten −24
.
Gabrielle is correct; Brittany multiplied the constants negative 6 times negative 4 and got plus 24, and she should have gotten negative 24.

Gabrielle is correct; Brittany did not use arrows to show the distributive property before she multiplied.
Gabrielle is correct; Brittany did not use arrows to show the distributive property before she multiplied.

Brittany is correct; Gabrielle multiplied the constants −6⋅−4
and got −24
, and she should have gotten +24
.
Brittany is correct; Gabrielle multiplied the constants negative 6 times negative 4 and got negative 24, and she should have gotten plus 24.

Brittany is correct; Gabrielle used arrows to show the distributive property.
Brittany is correct; Gabrielle used arrows to show the distributive property.
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page 10 of 10

To determine which student, Gabrielle or Brittany, is correct when simplifying the expression \((a - 6)(a - 4)\), let’s first perform the multiplication correctly using the distributive property (or FOIL method):

\[ (a - 6)(a - 4) \]

  1. Multiply \(a\) by both terms in the second parentheses:

    • \(a \cdot a = a^2\)
    • \(a \cdot (-4) = -4a\)

  2. Multiply \(-6\) by both terms in the second parentheses:

    • \(-6 \cdot a = -6a\)
    • \(-6 \cdot -4 = +24\)

Now, combine all these results: \[ a^2 - 4a - 6a + 24 \]

Combine like terms: \[ a^2 - 10a + 24 \]

So, the correct equivalent expression is \(a^2 - 10a + 24\).

Now, let’s analyze the options presented regarding Gabrielle and Brittany's work:

  • Option 1 states Gabrielle is correct and Brittany made an error with the multiplication \(-6 \cdot -4\) resulting in \(+24\) instead of \(-24\). This contradicts our correct multiplication because \(-6 \cdot -4\) is indeed \(+24\).

  • Option 2 is similar to Option 1 and repeats the same reasoning, which is also incorrect.

  • Option 3 mentions that Gabrielle is correct but criticizes Brittany for not showing the distributive property with arrows. This doesn't address the correctness of Brittany's answer, which is irrelevant to this context without knowing what Brittany's answer was.

  • Option 4 claims Brittany is correct and critiques Gabrielle's multiplication of constants as \(-24\), which is not true; Gabrielle’s multiplication was \(-6 \cdot -4 = +24\). This is also false.

  • Option 5 states Brittany is correct, which is untrue based on our multiplication.

  • Option 6 mentions Brittany being correct for showing the distributive property with arrows, but without more context, Gabrielle can also perform the steps correctly without using arrows.

Conclusion:

Based on the information provided and the calculations done, Gabrielle is correct; Brittany multiplied the constants \(-6 \cdot -4\) and got \(+24\), and she should have gotten \(-24\) is the correct evaluation if Brittany indeed misunderstood the signs in multiplication.

So, the correct choice is Option 1.