Question

multiply the polynomial expression (x+2)(x-4)
__x^2+__x+__

Answer 1

multiply to simplify the polynomial expression (r-1)(r^2-2r+3)
__r^3+__r^2+__r+__

Answer 2

simplify the polynomial expression (xy-2)(x^2+1)
__x^3y+__x^2+__xy+__

Answer 3

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

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

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

3. Multiply the inner terms:
\[
2 \cdot x = 2x
\]

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

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

Combine the like terms:
\[
x^2 - 2x - 8
\]

Thus, the final result is:
\[
x^2 - 2x - 8
\]

In the blanks, you would fill it as:
__1 for \(x^2\), __-2 for \(x\), and __-8 for the constant term.

So the result is:
- **First blank**: 1
- **Second blank**: -2
- **Third blank**: -8

Final answer: \(x^2 - 2x - 8\)

Answer 4

To multiply and simplify the polynomial expression \((r-1)(r^2-2r+3)\), we will distribute each term in the first polynomial to every term in the second polynomial.

1. **Multiply \(r\) by each term in \(r^2 - 2r + 3\):**
- \(r \cdot r^2 = r^3\)
- \(r \cdot (-2r) = -2r^2\)
- \(r \cdot 3 = 3r\)

2. **Multiply \(-1\) by each term in \(r^2 - 2r + 3\):**
- \(-1 \cdot r^2 = -r^2\)
- \(-1 \cdot (-2r) = 2r\)
- \(-1 \cdot 3 = -3\)

Now, let's combine all the results together:
\[
r^3 - 2r^2 + 3r - r^2 + 2r - 3
\]

Next, we will combine the like terms:
- For \(r^2\): \(-2r^2 - r^2 = -3r^2\)
- For \(r\): \(3r + 2r = 5r\)

So, the expression simplifies to:
\[
r^3 - 3r^2 + 5r - 3
\]

Finally, filling the given blanks:
- **First blank (for \(r^3\))**: 1
- **Second blank (for \(r^2\))**: -3
- **Third blank (for \(r\))**: 5
- **Fourth blank (for the constant)**: -3

Thus, the completed expression is:
\[
\mathbf{1}r^3 + \mathbf{-3}r^2 + \mathbf{5}r + \mathbf{-3}
\]

Answer 5

To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will distribute each term in the first polynomial to each term in the second polynomial.

1. **Multiply \(xy\) by each term in \(x^2 + 1\)**:
- \(xy \cdot x^2 = x^3y\)
- \(xy \cdot 1 = xy\)

2. **Multiply \(-2\) by each term in \(x^2 + 1\)**:
- \(-2 \cdot x^2 = -2x^2\)
- \(-2 \cdot 1 = -2\)

Now we combine all the results together:
\[
x^3y + xy - 2x^2 - 2
\]

Next, we can arrange the terms based on the powers of \(x\):
- \(x^3y\)
- \(-2x^2\)
- \(xy\)
- \(-2\)

Thus, the expression simplifies to:
\[
x^3y - 2x^2 + xy - 2
\]

Now, filling in the blanks:
- **First blank (for \(x^3y\))**: 1
- **Second blank (for \(x^2\))**: -2
- **Third blank (for \(xy\))**: 1
- **Fourth blank (for the constant)**: -2

So, the final answer is:
\[
\mathbf{1}x^3y + \mathbf{-2}x^2 + \mathbf{1}xy + \mathbf{-2}
\]