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

Question

.(2 points)
( )x^2+
( )x+ ( )

Answer 1

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

1. Multiply \(x\) by \(x\): \[ x \cdot x = x^2 \]

2. Multiply \(x\) by \(-2\): \[ x \cdot (-2) = -2x \]

3. Multiply \(8\) by \(x\): \[ 8 \cdot x = 8x \]

4. Multiply \(8\) by \(-2\): \[ 8 \cdot (-2) = -16 \]

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

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

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

Thus, filling in the blanks: \[ (1)x^2 + (6)x + (-16) \]

Answer 2

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

Answer 3

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

1. Multiply \(3x\) by \(x\): \[ 3x \cdot x = 3x^2 \]

2. Multiply \(3x\) by \(3\): \[ 3x \cdot 3 = 9x \]

Now, combine the results: \[ 3x(x + 3) = 3x^2 + 9x \]

Thus, the expression rewritten using the Distributive Property is: \[ 3x^2 + 9x \]

Answer 4

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

( )x^2 + ( )x + ( )

Answer 5

To find the product of the polynomials \((2x + 1)(x - 4)\) using the Distributive Property, we will multiply each term in the first polynomial by each term in the second polynomial.

1. Multiply \(2x\) by \(x\): \[ 2x \cdot x = 2x^2 \]

2. Multiply \(2x\) by \(-4\): \[ 2x \cdot (-4) = -8x \]

3. Multiply \(1\) by \(x\): \[ 1 \cdot x = x \]

4. Multiply \(1\) by \(-4\): \[ 1 \cdot (-4) = -4 \]

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

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

So, the product of the polynomials \((2x + 1)(x - 4)\) can be rewritten as: \[ 2x^2 + (-7)x + (-4) \]

Thus, filling in the blanks: \[ (2)x^2 + (-7)x + (-4) \]

Answer 6

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: x^2+2x−8




Option #1: (x+4)(x−2)
(Start)

Option #2: x^2−2x+4x−8

Option #3: x^2+2x−8

Option #4: x(x)−x(2)+4(x)−4(2)

(2 points)
First: Option #( )

Second: Option # ( )

Third: Option # ( )

Fourth: Option # ( )

Answer 7

To prove that Expression A \((x+4)(x−2)\) is equivalent to Expression B \(x^2+2x−8\), we can follow these steps using the distributive property.

1. First: Start with the original expression.

   • Option #1: \((x+4)(x−2)\)

2. Second: Apply the distributive property (also known as the FOIL method for binomials) to expand the expression.

   • Option #4: \(x(x)−x(2)+4(x)−4(2)\)

3. Third: Combine the like terms to simplify the expression.

   • Option #2: \(x^2−2x+4x−8\)

4. Fourth: Write the final simplified expression.

   • Option #3: \(x^2+2x−8\)

Putting it all together:

• First: Option # (1)
• Second: Option # (4)
• Third: Option # (2)
• Fourth: Option # (3)

Answer 8

Identify two expressions that will be equivalent to 2x^2−8x−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 # ( )

Answer 9

To identify two expressions that will be equivalent to \(2x^2 - 8x - 10\) when distributed, we can test each option by distributing and simplifying.

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

   • Distribute: \(2x(x) - 2x(4x) - 2x(5) = 2x^2 - 8x - 10\) (This simplifies to the original expression.)

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

   • Distribute: \((2x)(x) + (2)(x) + (2x)(-5) + (2)(-5) = 2x^2 + 2x - 10x - 10 = 2x^2 - 8x - 10\) (This simplifies to the original expression.)

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

   • Distribute \(2x(x - 5) \rightarrow 2x^2 - 10x\)
   • Distribute \(2(x - 5) \rightarrow 2x - 10\)
   • Combine: \(2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10\) (This simplifies to the original expression.)

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

   • Distribute: \(x^2 - 3x - 5x + 15 = x^2 - 8x + 15\) (This is not equivalent.)

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

   • This is not equivalent; it doesn't contain the \(2x^2\) term.

Conclusion: The expressions that are equivalent to \(2x^2 - 8x - 10\) when distributed are:

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

So, the answer is:

• Option # (1) and Option # (2).