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

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

Here’s how to do it step by step:

  1. Distribute \( x \) from the first parenthesis:

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

  2. Distribute \( 8 \) from the first parenthesis:

    • \( 8 \cdot x = 8x \)
    • \( 8 \cdot (-2) = -16 \)

Now, combine all the results together:

\[ x^2 - 2x + 8x - 16 \]

Next, combine the like terms \( -2x \) and \( 8x \):

\[ x^2 + 6x - 16 \]

Thus, the polynomial \( (x + 8)(x - 2) \) rewritten using the Distributive Property is

\[ \boxed{x^2 + 6x - 16}. \]

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

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

Here are the steps:

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

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

Now, combine the results:

\[ 3x(x + 3) = 3x^2 + 9x \]

Thus, the expression \( 3x(x + 3) \) rewritten using the Distributive Property is

\[ \boxed{3x^2 + 9x}. \]

Use the Distributive Property to rewrite the polynomial (x+8)(x−2).
x2+ x+

To rewrite the polynomial \( (x + 8)(x - 2) \) using the Distributive Property, we will distribute each term in the first expression to each term in the second expression. Here’s the step-by-step breakdown:

  1. Distribute \( x \) from the first parenthesis:

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

  2. Distribute \( 8 \) from the first parenthesis:

    • \( 8 \cdot x = 8x \)
    • \( 8 \cdot (-2) = -16 \)

Now, let's combine all these results together:

\[ x^2 - 2x + 8x - 16 \]

Now, combine the like terms \( -2x \) and \( 8x \):

\[ x^2 + 6x - 16 \]

So the polynomial \( (x + 8)(x - 2) \) rewritten using the Distributive Property is

\[ \boxed{x^2 + 6x - 16}. \]

Find the product of the polynomials (2x+1)(x−4)

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 the step-by-step process:

  1. First: Multiply the first terms: \[ 2x \cdot x = 2x^2 \]

  2. Outer: Multiply the outer terms: \[ 2x \cdot (-4) = -8x \]

  3. Inner: Multiply the inner terms: \[ 1 \cdot x = 1x = x \]

  4. Last: Multiply the last terms: \[ 1 \cdot (-4) = -4 \]

Now, combine all of these results together: \[ 2x^2 - 8x + x - 4 \]

Next, combine the like terms \( -8x \) and \( x \): \[ 2x^2 - 7x - 4 \]

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

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)

To prove that Expression A \( (x+4)(x−2) \) is equivalent to Expression B \( x^2 + 2x - 8 \), we can organize the steps in the correct order. Here's how it should be arranged:

  1. Option #1: \( (x + 4)(x - 2) \) (Start)
  2. Option #4: \( x(x) - x(2) + 4(x) - 4(2) \) (Applying the Distributive Property)
  3. Option #2: \( x^2 - 2x + 4x - 8 \) (Combining all the distributed terms)
  4. Option #3: \( x^2 + 2x - 8 \) (Combining like terms)

So, the ordered steps to prove that Expression A is equivalent to Expression B are:

  1. Option #1
  2. Option #4
  3. Option #2
  4. Option #3

This sequence will successfully show that \( (x + 4)(x - 2) = x^2 + 2x - 8 \).

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

To identify two expressions that will be equivalent to \( 2x^2 - 8x - 10 \) when distributed, let's check each option by distributing them and comparing the results.

  1. Option #1: \( 2x(x - 4x - 5) \)
    This option seems incorrectly formatted. However, if interpreted as \( 2x((x - 4) - 5) \) or any logical similar expression, it needs clarification.

  2. Option #2: \( (2x + 2)(x - 5) \)
    Distributing this gives: \[ 2x \cdot x + 2 \cdot x - 2x \cdot 5 - 2 \cdot 5 = 2x^2 + 2x - 10x - 10 = 2x^2 - 8x - 10 \] This matches!

  3. Option #3: \( 2x(x - 5) + 2(x - 5) \)
    This option can be simplified as follows: \[ 2x(x - 5) + 2(x - 5) = (2x + 2)(x - 5) \quad \text{(Factoring out } (x - 5) \text{)} \] Since we already know that \( (2x + 2)(x - 5) = 2x^2 - 8x - 10 \) from Option #2, this option also corresponds.

  4. Option #4: \( (x - 5)(x - 3) \)
    Distributing gives: \[ x \cdot x - 3x - 5x + 15 = x^2 - 8x + 15 \quad \text{(This does not match)} \]

  5. Option #5: \( -6x - 10 \)
    This is clearly different from \( 2x^2 - 8x - 10 \).

Conclusion:

The two options that are equivalent to \( 2x^2 - 8x - 10 \) when distributed are Option #2 and Option #3.

To order them with the lesser option first:

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

Thus, the final answer would be:

Option #2 followed by Option #3.