To find the product of the polynomials \( (9x^2 - 6x + 1)(3x - 1) \), we can use the distributive property (also known as the distributive law of multiplication over addition).
Let's distribute \( (3x - 1) \) to each term in \( (9x^2 - 6x + 1) \):
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Distributing \( 3x \): \[ 3x \cdot 9x^2 = 27x^3 \] \[ 3x \cdot (-6x) = -18x^2 \] \[ 3x \cdot 1 = 3x \]
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Distributing \( -1 \): \[ -1 \cdot 9x^2 = -9x^2 \] \[ -1 \cdot (-6x) = 6x \] \[ -1 \cdot 1 = -1 \]
Now, we can combine all these terms together: \[ 27x^3 + (-18x^2 - 9x^2) + (3x + 6x) - 1 \] This simplifies to: \[ 27x^3 - 27x^2 + 9x - 1 \]
The final product of the polynomials \( (9x^2 - 6x + 1)(3x - 1) \) is: \[ 27x^3 - 27x^2 + 9x - 1 \]
Therefore, the correct response is: \[ 27x^3 - 27x^2 + 9x - 1 \]
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