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Use the Distributive Property to rewrite the polynomial (x+8)(x−2) .(2 points)

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9 answers

To use the Distributive Property to rewrite the polynomial \( (x+8)(x-2) \), we will distribute each term in the first parenthesis \( (x + 8) \) to each term in the second parenthesis \( (x - 2) \).

Here’s how it works step by step:

  1. Distribute \( x \) in \( (x + 8) \): \[ x \cdot x = x^2 \] \[ x \cdot (-2) = -2x \]

  2. Distribute \( 8 \) in \( (x + 8) \): \[ 8 \cdot x = 8x \] \[ 8 \cdot (-2) = -16 \]

  3. Now, combine all the results: \[ x^2 - 2x + 8x - 16 \]

  4. Combine the like terms \(-2x\) and \(8x\): \[ x^2 + 6x - 16 \]

Thus, the polynomial \( (x+8)(x-2) \) can be rewritten as: \[ x^2 + 6x - 16 \]

Rewrite the expression 3x(x + 3) using the Distributive Property.(1 point)

To rewrite the expression \( 3x(x + 3) \) using the Distributive Property, you will distribute \( 3x \) to each term inside the parentheses \( (x + 3) \).

Here are the steps:

  1. Distribute \( 3x \) to \( x \): \[ 3x \cdot x = 3x^2 \]

  2. Distribute \( 3x \) to \( 3 \): \[ 3x \cdot 3 = 9x \]

  3. Combine the results: \[ 3x(x + 3) = 3x^2 + 9x \]

So, the expression \( 3x(x + 3) \) can be rewritten as: \[ 3x^2 + 9x \]

Find the product of the polynomials (2x+1)(x−4) .(2 points)
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Remaining Attempts : 3

To find the product of the polynomials \( (2x+1)(x-4) \), we will use the Distributive Property (also known as the FOIL method for binomials).

Here's how to do it step by step:

  1. Distribute \(2x\):

    • \(2x \cdot x = 2x^2\)
    • \(2x \cdot (-4) = -8x\)

  2. Distribute \(1\):

    • \(1 \cdot x = x\)
    • \(1 \cdot (-4) = -4\)

  3. Combine all the results: \[ 2x^2 - 8x + x - 4 \]

  4. Combine the like terms \(-8x\) and \(x\): \[ 2x^2 - 7x - 4 \]

Thus, the product of the polynomials \( (2x+1)(x-4) \) is: \[ 2x^2 - 7x - 4 \]

So, if you wish to fill in the blank:

  • \(x^2\) coefficient: 2
  • \(x\) coefficient: -7
  • Constant term: -4
From first to last, order the steps to prove that expression A is equivalent to expression B.

Expression A: (x+4)(x−2)

Expression B: x2+2x−8



Option #1: (x+4)(x−2) (Start)

Option #2: x2−2x+4x−8

Option #3: x2+2x−8

Option #4: x(x)−x(2)+4(x)−4(2)

(2 points)
First: Option #

Second: Option #

Third: Option #

Fourth: Option #

To prove that Expression A \((x + 4)(x - 2)\) is equivalent to Expression B \(x^2 + 2x - 8\), we need to follow the steps of using the Distributive Property (or FOIL method). Here’s the correct order:

  1. First: Option #1: \((x + 4)(x - 2)\) (Start)

  2. Second: Option #4: \(x(x) - x(2) + 4(x) - 4(2)\) (Distributing each term)

  3. Third: Option #2: \(x^2 - 2x + 4x - 8\) (Combine the products)

  4. Fourth: Option #3: \(x^2 + 2x - 8\) (Combine like terms)

So, the ordering is:

  • First: Option #1
  • Second: Option #4
  • Third: Option #2
  • Fourth: Option #3
Identify two expressions that will be equivalent to 2x2−8x−10 when distributed. Enter the option with the lesser number first.

Option #1: 2x(x−4x−5)

Option #2: (2x+2)(x−5)

Option #3: 2x(x−5)+2(x−5)

Option #4: (x−5)(x−3)

Option #5: −6x−10

(1 point)
Option #
and Option #

To identify which expressions are equivalent to \(2x^2 - 8x - 10\) when distributed, let's simplify each of the given options.

Option #1: \(2x(x-4x-5)\)

This option looks like it contains a typo (two variables after \(4\)). Assuming it means \(2x(x - 4)(x - 5)\):

  • Distribution: \(2x[x - (4x + 5)] = 2x(x - 4x - 5)\) is not correct in structure as it overcomplicates.

Option #2: \((2x + 2)(x - 5)\)

  • Distribution: \[ (2x + 2)(x - 5) = 2x \cdot x + 2x \cdot (-5) + 2 \cdot x + 2 \cdot (-5) \] \[ = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \]
  • This expression is equivalent to \(2x^2 - 8x - 10\).

Option #3: \(2x(x - 5) + 2(x - 5)\)

  • Distribution: \[ 2x(x - 5) + 2(x - 5) = 2x^2 - 10x + 2x - 10 \] \[ = 2x^2 - 8x - 10 \]
  • This expression is equivalent to \(2x^2 - 8x - 10\).

Option #4: \((x - 5)(x - 3)\)

  • Distribution: \[ (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
  • This is not equivalent to \(2x^2 - 8x - 10\).

Option #5: \(-6x - 10\)

  • This is not a polynomial in \(x^2\), hence not equivalent.

Summary:

The equivalent expressions are:

  • Option #2: \((2x + 2)(x - 5)\)
  • Option #3: \(2x(x - 5) + 2(x - 5)\)

In order of lesser number first:

  • Option #2
  • Option #3

So the answer is: Option #2 and Option #3.