To find the product of the polynomials \( (9x^2 - 6x + 1)(3x - 1) \), we can apply the distributive property (also known as the FOIL method for binomials).
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Multiply \( 9x^2 \) by each term in \( (3x - 1) \): \[ 9x^2 \cdot 3x = 27x^3 \] \[ 9x^2 \cdot (-1) = -9x^2 \]
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Multiply \( -6x \) by each term in \( (3x - 1) \): \[ -6x \cdot 3x = -18x^2 \] \[ -6x \cdot (-1) = 6x \]
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Multiply \( 1 \) by each term in \( (3x - 1) \): \[ 1 \cdot 3x = 3x \] \[ 1 \cdot (-1) = -1 \]
Now, combine all these results: \[ 27x^3 + (-9x^2 - 18x^2) + (6x + 3x) + (-1) \] Combine the like terms: \[ 27x^3 + (-27x^2) + 9x - 1 \]
This simplifies to: \[ 27x^3 - 27x^2 + 9x - 1 \]
Thus, the product of the polynomials is: \[ \boxed{27x^3 - 27x^2 + 9x - 1} \]
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