To find \((f \cdot g)(x)\) (which means \(f(x) \cdot g(x)\)), we need to multiply the functions \(f(x)\) and \(g(x)\) together.
You provided:
- \(f(x) = x^2 + 8x + 15\)
- \(g(x) = x - 9\)
Now we will multiply these two expressions:
\[ (f \cdot g)(x) = (x^2 + 8x + 15)(x - 9) \]
Using the distributive property (also known as the FOIL method for binomials), we can expand this expression:
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Multiply \(x^2\) by both terms in \(g(x)\): \[ x^2 \cdot x = x^3 \] \[ x^2 \cdot (-9) = -9x^2 \]
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Multiply \(8x\) by both terms in \(g(x)\): \[ 8x \cdot x = 8x^2 \] \[ 8x \cdot (-9) = -72x \]
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Multiply \(15\) by both terms in \(g(x)\): \[ 15 \cdot x = 15x \] \[ 15 \cdot (-9) = -135 \]
Now, combine all these terms together: \[ x^3 + (-9x^2 + 8x^2) + (-72x + 15x) - 135 \]
Simplifying the coefficients: \[ x^3 + (-1x^2) + (-57x) - 135 \]
So we have: \[ (f \cdot g)(x) = x^3 - x^2 - 57x - 135 \]
Therefore, the final answer for \((f \cdot g)(x)\) is: \[ \boxed{x^3 - x^2 - 57x - 135} \]