Question

expand the expression to a polynomial in standard form

(2x^2+x+3)(3x^2-2x+6) and does it have parenthenses around it in standard form


you made me get the other question wrong so dont make me fail

Answer 1

To expand the expression \((2x^2 + x + 3)(3x^2 - 2x + 6)\) and write it in standard form, we will use the distributive property (also known as the FOIL method for binomials, but here we apply it to polynomials).

1. **Distribute each term in the first polynomial to every term in the second polynomial**:
- \(2x^2 \cdot 3x^2\)
- \(2x^2 \cdot (-2x)\)
- \(2x^2 \cdot 6\)
- \(x \cdot 3x^2\)
- \(x \cdot (-2x)\)
- \(x \cdot 6\)
- \(3 \cdot 3x^2\)
- \(3 \cdot (-2x)\)
- \(3 \cdot 6\)

2. **Calculate each of these products**:
- \(2x^2 \cdot 3x^2 = 6x^4\)
- \(2x^2 \cdot (-2x) = -4x^3\)
- \(2x^2 \cdot 6 = 12x^2\)
- \(x \cdot 3x^2 = 3x^3\)
- \(x \cdot (-2x) = -2x^2\)
- \(x \cdot 6 = 6x\)
- \(3 \cdot 3x^2 = 9x^2\)
- \(3 \cdot (-2x) = -6x\)
- \(3 \cdot 6 = 18\)

3. **Combine the results**:
- \(6x^4\)
- Combine like terms for \(x^3\): \(-4x^3 + 3x^3 = -x^3\)
- Combine like terms for \(x^2\): \(12x^2 - 2x^2 + 9x^2 = 19x^2\)
- Combine like terms for \(x\): \(6x - 6x = 0\)
- The constant term is \(18\)

4. **Write the final polynomial in standard form**:
\[
\boxed{6x^4 - x^3 + 19x^2 + 18}
\]

To clarify, the final answer is indeed a polynomial in standard form (which arranges the terms in descending order of their degrees) without parentheses around it, as standard form typically does not include parentheses.