Questions: 1. Evaluate Expression: Given that a=3, b=5, and c=2, evaluate the following expression: (a^3 * b^2 + 4(c^2 - 1)) / (6(c^2 + a/3)) - 7/c 2. Simplify the following polynomial expression: (3 x^2 - 2 x + 4) - (2 x + 1)(8 x - 1)

1. Evaluate Expression: Given that a=3, b=5, and c=2, evaluate the following expression: (a^3 * b^2 + 4(c^2 - 1)) / (6(c^2 + a/3)) - 7/c
2. Simplify the following polynomial expression:
(3 x^2 - 2 x + 4) - (2 x + 1)(8 x - 1)

Transcript text: 1. Evaluate Expression: Given that $a=3, b=5$, and $c=2$, evaluate the following expression: $\frac{\left(a^{3} \cdot b^{2}\right)+4\left(c^{2}-1\right)}{6\left(c^{2}+\frac{a}{3}\right)}-\frac{7}{c}$ 2. Simplify the following polynomial expression: \[ \left(3 x^{2}-2 x+4\right)-(2 x+1)(8 x-1) \]

Solution

Solution Steps

Evaluate Expression: Given the values of $a$, $b$, and $c$, substitute them into the expression and perform the arithmetic operations step by step.
Simplify Polynomial: Expand the polynomial expression by distributing the terms and then combine like terms.

Step 1: Evaluate the Expression

Given the values $ a = 3 $, $ b = 5 $, and $ c = 2 $, we substitute these into the expression:

\[ \frac{(a^{3} \cdot b^{2}) + 4(c^{2} - 1)}{6(c^{2} + \frac{a}{3})} - \frac{7}{c} \]

Calculating the numerator:

\[ a^{3} \cdot b^{2} = 3^{3} \cdot 5^{2} = 27 \cdot 25 = 675 \] \[ 4(c^{2} - 1) = 4(2^{2} - 1) = 4(4 - 1) = 4 \cdot 3 = 12 \] Thus, the numerator becomes:

\[ 675 + 12 = 687 \]

Now, calculating the denominator:

\[ 6(c^{2} + \frac{a}{3}) = 6(2^{2} + \frac{3}{3}) = 6(4 + 1) = 6 \cdot 5 = 30 \]

Now we can evaluate the entire expression:

\[ \frac{687}{30} - \frac{7}{2} = 22.9 - 3.5 = 19.4 \]

Step 2: Simplify the Polynomial Expression

We simplify the polynomial expression:

\[ (3x^{2} - 2x + 4) - (2x + 1)(8x - 1) \]

First, we expand $ (2x + 1)(8x - 1) $:

\[ (2x + 1)(8x - 1) = 16x^{2} - 2x + 8x - 1 = 16x^{2} + 6x - 1 \]

Now substituting back into the expression:

\[ 3x^{2} - 2x + 4 - (16x^{2} + 6x - 1) \]