Question

Expand the expression to a polynomial in standard form:
left parenthesis, 3, x, squared, minus, x, plus, 3, right parenthesis, left parenthesis, 3, x, squared, plus, x, minus, 2, right parenthesis
(3x
2
−x+3)(3x
2
+x−2)

Answer 1

To expand the expression \((3x^2 - x + 3)(3x^2 + x - 2)\), we will use the distributive property (also known as the FOIL method for binomials). We'll multiply each term from the first polynomial by each term from the second polynomial.

1. **Distributing \(3x^2\):**
- \(3x^2 \cdot 3x^2 = 9x^4\)
- \(3x^2 \cdot x = 3x^3\)
- \(3x^2 \cdot -2 = -6x^2\)

2. **Distributing \(-x\):**
- \(-x \cdot 3x^2 = -3x^3\)
- \(-x \cdot x = -x^2\)
- \(-x \cdot -2 = 2x\)

3. **Distributing \(3\):**
- \(3 \cdot 3x^2 = 9x^2\)
- \(3 \cdot x = 3x\)
- \(3 \cdot -2 = -6\)

Now, we will combine all these results:

\[
9x^4 + (3x^3 - 3x^3) + (-6x^2 - x^2 + 9x^2) + (2x + 3x) - 6
\]

Next, we combine like terms:

- For \(x^4\) terms: \(9x^4\)
- For \(x^3\) terms: \(3x^3 - 3x^3 = 0\)
- For \(x^2\) terms: \(-6x^2 - x^2 + 9x^2 = 2x^2\)
- For \(x\) terms: \(2x + 3x = 5x\)
- The constant term: \(-6\)

Now we can write the expanded polynomial:

\[
9x^4 + 2x^2 + 5x - 6
\]

So, the final answer in standard form is:

\[
\boxed{9x^4 + 2x^2 + 5x - 6}
\]