Multiply the polynomial expression (x+2)(x−4)

Question

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

Answer 1

To multiply the polynomial expression \((x+2)(x-4)\), we can use the distributive property (also known as the FOIL method for binomials).

\[ (x + 2)(x - 4) = x(x) + x(-4) + 2(x) + 2(-4) \]

Now, let's simplify that step by step:

1. \(x \cdot x = x^2\)
2. \(x \cdot (-4) = -4x\)
3. \(2 \cdot x = 2x\)
4. \(2 \cdot (-4) = -8\)

Now, combine all these:

\[ x^2 - 4x + 2x - 8 \]

Next, combine like terms:

\(-4x + 2x = -2x\)

So, we have:

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

Thus, the final expression is:

\[ 1x^2 + (-2)x + (-8) \]

So, filling in the blanks:

\((1)x^2 + (-2)x + (-8)\).

Answer 2

Multiply to simplify the polynomial expression (r−1)(r^2−2r+3)
.(2 points)
( )r^3 + ( )r^2+ ( )r + ( )

Answer 3

To multiply and simplify the polynomial expression \((r-1)(r^2-2r+3)\), we can use the distributive property.

We will distribute \(r - 1\) to each term in \(r^2 - 2r + 3\):

1. Distributing \(r\): \[ r \cdot (r^2) = r^3 \] \[ r \cdot (-2r) = -2r^2 \] \[ r \cdot (3) = 3r \]

2. Distributing \(-1\): \[ -1 \cdot (r^2) = -r^2 \] \[ -1 \cdot (-2r) = 2r \] \[ -1 \cdot (3) = -3 \]

Now, let's combine all the terms:

\[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Now, combine like terms:

• The \(r^2\) terms: \(-2r^2 - r^2 = -3r^2\)
• The \(r\) terms: \(3r + 2r = 5r\)

Putting it all together, we get:

\[ r^3 - 3r^2 + 5r - 3 \]

Thus, the final simplified expression is:

\[ (1)r^3 + (-3)r^2 + (5)r + (-3) \]

So, filling in the blanks:

\((1)r^3 + (-3)r^2 + (5)r + (-3)\).

Answer 4

Simplify the polynomial expression (xy−2)(x^2+1)

( )x^3y + ( )x^2 + ( )xy + ( )

Answer 5

To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will use the distributive property.

We will distribute the terms of the first polynomial \(xy - 2\) to each term in the second polynomial \(x^2 + 1\):

1. Distributing \(xy\):

   • \(xy \cdot x^2 = xyx^2 = x^3y\)
   • \(xy \cdot 1 = xy\)

2. Distributing \(-2\):

   • \(-2 \cdot x^2 = -2x^2\)
   • \(-2 \cdot 1 = -2\)

Now, combine all these results together:

\[ x^3y - 2x^2 + xy - 2 \]

Now, let's rearrange and group the terms:

\[ x^3y + xy - 2x^2 - 2 \]

Thus, the simplified expression is:

\[ (1)x^3y + (-2)x^2 + (1)xy + (-2) \]

So, filling in the blanks:

\((1)x^3y + (-2)x^2 + (1)xy + (-2)\).

Answer 6

Which of the following shows that polynomials form a closed system under multiplication?

Option #1: 3(2/x+5)=6/x+15

Option #2: 8x+7=5x+3x+1+6

Option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24

Option #4: 2x^1/2⋅3=6x−−√x

(1 point)
Option #

Answer 7

To determine which option shows that polynomials form a closed system under multiplication, we need to recall that a set is closed under an operation if applying that operation to members of the set always produces a member of the set. In this case, we want to check if multiplying two polynomials results in a polynomial.

Let's analyze each option:

1. Option #1: \(3\left(\frac{2}{x}+5\right) = \frac{6}{x} + 15\)

   • This is not an example of polynomial multiplication, as it involves a rational expression (\(\frac{2}{x}\)), which is not a polynomial.

2. Option #2: \(8x + 7 = 5x + 3x + 1 + 6\)

   • This is an expression showing addition and equating two polynomial expressions. It does not demonstrate multiplication.

3. Option #3: \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)

   • This shows the product of two polynomials \((2x^2 - 4)\) and \((3y + 6)\), which results in another polynomial. Therefore, this option demonstrates that polynomials are closed under multiplication.

4. Option #4: \(2x^{1/2} \cdot 3 = 6\sqrt{x}\)

   • This involves a term \(x^{1/2}\) which is a radical expression, and thus it is not considered a polynomial.

Based on this analysis, the correct option that shows that polynomials form a closed system under multiplication is:

Option #3.