Question

Use the following multiplication problem to show how multiplying polynomials is a closed system.

(x2−1)(−2x+5)




say it like how a 8th grader would and easy to copy and paste for me

Answer 1

Sure! Let's break it down step by step.

When we multiply two polynomials, we're using a closed system, which means that when we multiply them, we only get other polynomials as the result.

Let's look at the problem \((x^2 - 1)(-2x + 5)\):

**Step 1: Distribute Each Term**
We will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

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

2. **Multiply \(x^2\) by \(5\):**
\[
x^2 \cdot 5 = 5x^2
\]

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

4. **Multiply \(-1\) by \(5\):**
\[
-1 \cdot 5 = -5
\]

**Step 2: Combine All the Results**
Now we put all the results together:
\[
-2x^3 + 5x^2 + 2x - 5
\]

**Step 3: Write the Final Polynomial**
So the final result of \((x^2 - 1)(-2x + 5)\) is:
\[
-2x^3 + 5x^2 + 2x - 5
\]

**Conclusion:**
Since \(-2x^3 + 5x^2 + 2x - 5\) is also a polynomial, this shows that when we multiply two polynomials, we always get another polynomial as the result. That's why we say multiplying polynomials is a "closed system!"

And that's it! Multiplying polynomials just gives us more polynomials!