Question

Use distributive property to find the product of (2𝑥 2 + 3𝑥 + 2)(5𝑥 2 − 10𝑥 + 3) (write the solution out)

Answer 1

To find the product of the polynomials \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) using the distributive property, we will distribute each term in the first polynomial to each term in the second polynomial.

Let's break this down step by step.

1. **Distribute \(2x^2\)**:
\[
2x^2 \cdot (5x^2 - 10x + 3) = 2x^2 \cdot 5x^2 + 2x^2 \cdot (-10x) + 2x^2 \cdot 3
\]
\[
= 10x^4 - 20x^3 + 6x^2
\]

2. **Distribute \(3x\)**:
\[
3x \cdot (5x^2 - 10x + 3) = 3x \cdot 5x^2 + 3x \cdot (-10x) + 3x \cdot 3
\]
\[
= 15x^3 - 30x^2 + 9x
\]

3. **Distribute \(2\)**:
\[
2 \cdot (5x^2 - 10x + 3) = 2 \cdot 5x^2 + 2 \cdot (-10x) + 2 \cdot 3
\]
\[
= 10x^2 - 20x + 6
\]

4. **Combine all the results**:
Now, we will combine all the terms we have found:
\[
(10x^4 - 20x^3 + 6x^2) + (15x^3 - 30x^2 + 9x) + (10x^2 - 20x + 6)
\]

5. **Combine like terms**:
- For \(x^4\): \(10x^4\)
- For \(x^3\): \(-20x^3 + 15x^3 = -5x^3\)
- For \(x^2\): \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
- For \(x\): \(9x - 20x = -11x\)
- Constant term: \(6\)

Thus, the final result combining all these terms is:
\[
\boxed{10x^4 - 5x^3 - 14x^2 - 11x + 6}
\]

Answer 2

is that the product

Answer 3

Yes, the expression \(10x^4 - 5x^3 - 14x^2 - 11x + 6\) is the product of the polynomials \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) obtained using the distributive property.

To summarize, here's how we arrived at that final expression:

1. **Distributing each term**:
- \(2x^2\) contributed \(10x^4 - 20x^3 + 6x^2\)
- \(3x\) contributed \(15x^3 - 30x^2 + 9x\)
- \(2\) contributed \(10x^2 - 20x + 6\)

2. **Combining like terms**:
- \(x^4\) terms: \(10x^4\)
- \(x^3\) terms: \(-20x^3 + 15x^3 = -5x^3\)
- \(x^2\) terms: \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
- \(x\) terms: \(9x - 20x = -11x\)
- Constant term: \(6\)

Putting it all together gives:
\[
10x^4 - 5x^3 - 14x^2 - 11x + 6
\]

This is indeed the correct product of the two polynomials.