To expand the expression \((2x^2+x+3)(3x^2-2x+6)\), we will apply the distributive property (also known as the FOIL method when multiplying two binomials) to multiply each term in the first polynomial by each term in the second polynomial.
Let's break this down:
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Multiply \(2x^2\) by each term in \(3x^2 - 2x + 6\):
- \(2x^2 \cdot 3x^2 = 6x^4\)
- \(2x^2 \cdot (-2x) = -4x^3\)
- \(2x^2 \cdot 6 = 12x^2\)
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Multiply \(x\) by each term in \(3x^2 - 2x + 6\):
- \(x \cdot 3x^2 = 3x^3\)
- \(x \cdot (-2x) = -2x^2\)
- \(x \cdot 6 = 6x\)
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Multiply \(3\) by each term in \(3x^2 - 2x + 6\):
- \(3 \cdot 3x^2 = 9x^2\)
- \(3 \cdot (-2x) = -6x\)
- \(3 \cdot 6 = 18\)
Now, let's combine all the results together:
\[ 6x^4 + (-4x^3 + 3x^3) + (12x^2 - 2x^2 + 9x^2) + (6x - 6x) + 18 \]
Now, we simplify each group by combining like terms:
- For \(x^4\): \(6x^4\)
- For \(x^3\): \(-4x^3 + 3x^3 = -x^3\)
- For \(x^2\): \(12x^2 - 2x^2 + 9x^2 = 19x^2\)
- For \(x\): \(6x - 6x = 0\)
- The constant term remains: \(18\)
Putting it all together in standard form (from highest degree to lowest degree):
\[ 6x^4 - x^3 + 19x^2 + 18 \]
Answer:
The expanded polynomial is: \[ 6x^4 - x^3 + 19x^2 + 18 \]
And regarding your questions:
- Does it have parentheses around it in standard form?: No, in standard form, we do not include parentheses.
- What does "boxed" mean?: In some contexts, "boxed" refers to writing an answer in a box (visually enclosing it to highlight it), but it may depend on your instructor's preferences or the specific task you are given.
If you have any further questions or need clarifications, please feel free to ask!
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