To expand the expression \((3x + 2)(3x^2 - 3x - 7)\), you can use the distributive property (also known as the FOIL method for binomials).
Here’s how to do it step by step:
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Distribute \(3x\) to each term in the second polynomial: \[ 3x \cdot 3x^2 = 9x^3 \] \[ 3x \cdot (-3x) = -9x^2 \] \[ 3x \cdot (-7) = -21x \]
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Now distribute \(2\) to each term in the second polynomial: \[ 2 \cdot 3x^2 = 6x^2 \] \[ 2 \cdot (-3x) = -6x \] \[ 2 \cdot (-7) = -14 \]
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Combine all these results together: \[ 9x^3 - 9x^2 - 21x + 6x^2 - 6x - 14 \]
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Now, combine like terms:
- For \(x^2\) terms: \(-9x^2 + 6x^2 = -3x^2\)
- For \(x\) terms: \(-21x - 6x = -27x\)
So, the expression simplifies to: \[ 9x^3 - 3x^2 - 27x - 14 \]
Thus, the final answer in polynomial standard form is: \[ \boxed{9x^3 - 3x^2 - 27x - 14} \]
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