Let's break down your request step by step.
First, we'll multiply the polynomial \((b + 8)(3b - 6)\):
- Using the distributive property (also known as the FOIL method for polynomials):
- First: \(b \cdot 3b = 3b^2\)
- Outside: \(b \cdot -6 = -6b\)
- Inside: \(8 \cdot 3b = 24b\)
- Last: \(8 \cdot -6 = -48\)
Now combining these results: \[ 3b^2 + 24b - 6b - 48 \] Combine like terms: \[ 3b^2 + (24b - 6b) - 48 = 3b^2 + 18b - 48 \]
So, \((b + 8)(3b - 6) = 3b^2 + 18b - 48\).
Next, let's simplify the different expressions you provided:
-
Expression 1: \(6^2 + 185 - 36\) \[ 6^2 = 36 \implies 36 + 185 - 36 = 185 \]
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Expression 2: \(30^2 + 300 - 48\) \[ 30^2 = 900 \implies 900 + 300 - 48 = 1152 \]
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Expression 3: \(\frac{3b^2}{18} \cdot b - 48\) \[ \frac{3b^2}{18} = \frac{1}{6}b^2 \implies \frac{1}{6}b^2 \cdot b - 48 = \frac{1}{6}b^3 - 48 \]
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Expression 4: \(6^2 + 306 - 48\) \[ 6^2 = 36 \implies 36 + 306 - 48 = 294 \]
To summarize the results:
- The polynomial multiplication gives: \(3b^2 + 18b - 48\)
- Expressions evaluated:
- \(6^2 + 185 - 36 = 185\)
- \(30^2 + 300 - 48 = 1152\)
- \(\frac{3b^2}{18} \cdot b - 48 = \frac{1}{6}b^3 - 48\)
- \(6^2 + 306 - 48 = 294\)