PRODUCT EXPRESSIONS AND SUM EXPRESSIONS

The process of writing product expressions as sum expressions is called expansion. It is sometimes also referred to as the multiplication of algebraic expressions.

1. (a) Complete the table for the given values of [tex]$x, y,$[/tex] and [tex][tex]$z$[/tex]:[/tex]

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & $3(x+2y+4z)$ & $3x+6y+12z$ & $3x+2y+4z$ \\
\hline
\begin{tabular}{l}
$x=1$ \\
$y=2$ \\
$z=3$
\end{tabular} & & & \\
\hline
\begin{tabular}{l}
$x=10$ \\
$y=20$ \\
$z=30$
\end{tabular} & & & \\
\hline
\begin{tabular}{l}
$x=23$ \\
$y=60$ \\
$z=10$
\end{tabular} & & & \\
\hline
\begin{tabular}{l}
$x=14$ \\
$y=0$ \\
$z=1$
\end{tabular} & & & \\
\hline
\begin{tabular}{l}
$x=5$ \\
$y=9$ \\
$z=32$
\end{tabular} & & & \\
\hline
\end{tabular}
\][/tex]

(b) Which sum expression and product expression are equivalent?

Sure! Let's go through the problem step-by-step and fill in the values for the expressions, and then identify the equivalent product and sum expressions.

### Step-by-Step Solution:

Step 1: Calculate the expressions for each set of values
We need to calculate three expressions for each given set of values of [tex]$ x, y, $[/tex] and [tex]$ z $[/tex]:
- [tex]$ 3(x + 2y + 4z) $[/tex]
- [tex]$ 3x + 6y + 12z $[/tex]
- [tex]$ 3x + 2y + 4z $[/tex]

Given values:
1. [tex]$ x = 1 $[/tex], [tex]$ y = 2 $[/tex], [tex]$ z = 3 $[/tex]
2. [tex]$ x = 10 $[/tex], [tex]$ y = 20 $[/tex], [tex]$ z = 30 $[/tex]
3. [tex]$ x = 23 $[/tex], [tex]$ y = 60 $[/tex], [tex]$ z = 10 $[/tex]
4. [tex]$ x = 14 $[/tex], [tex]$ y = 0 $[/tex], [tex]$ z = 1 $[/tex]
5. [tex]$ x = 5 $[/tex], [tex]$ y = 9 $[/tex], [tex]$ z = 32 $[/tex]

Using these values, let's calculate the expressions:

1. For [tex]$ x = 1 $[/tex], [tex]$ y = 2 $[/tex], [tex]$ z = 3 $[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(1 + 2(2) + 4(3)) = 3(1 + 4 + 12) = 3 \times 17 = 51 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(1) + 6(2) + 12(3) = 3 + 12 + 36 = 51 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(1) + 2(2) + 4(3) = 3 + 4 + 12 = 19 \][/tex]

2. For [tex]$ x = 10 $[/tex], [tex]$ y = 20 $[/tex], [tex]$ z = 30 $[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(10 + 2(20) + 4(30)) = 3(10 + 40 + 120) = 3 \times 170 = 510 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(10) + 6(20) + 12(30) = 30 + 120 + 360 = 510 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(10) + 2(20) + 4(30) = 30 + 40 + 120 = 190 \][/tex]

3. For [tex]$ x = 23 $[/tex], [tex]$ y = 60 $[/tex], [tex]$ z = 10 $[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(23 + 2(60) + 4(10)) = 3(23 + 120 + 40) = 3 \times 183 = 549 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(23) + 6(60) + 12(10) = 69 + 360 + 120 = 549 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(23) + 2(60) + 4(10) = 69 + 120 + 40 = 229 \][/tex]

4. For [tex]$ x = 14 $[/tex], [tex]$ y = 0 $[/tex], [tex]$ z = 1 $[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(14 + 2(0) + 4(1)) = 3(14 + 0 + 4) = 3 \times 18 = 54 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(14) + 6(0) + 12(1) = 42 + 0 + 12 = 54 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(14) + 2(0) + 4(1) = 42 + 0 + 4 = 46 \][/tex]

5. For [tex]$ x = 5 $[/tex], [tex]$ y = 9 $[/tex], [tex]$ z = 32 $[/tex]:
[tex]\[ 3(x + 2y + 4z) = 3(5 + 2(9) + 4(32)) = 3(5 + 18 + 128) = 3 \times 151 = 453 \][/tex]
[tex]\[ 3x + 6y + 12z = 3(5) + 6(9) + 12(32) = 15 + 54 + 384 = 453 \][/tex]
[tex]\[ 3x + 2y + 4z = 3(5) + 2(9) + 4(32) = 15 + 18 + 128 = 161 \][/tex]

### Step 2: Fill in the Table

[tex]\[ \begin{array}{|c|c|c|c|} \hline & 3(x + 2y + 4z) & 3x + 6y + 12z & 3x + 2y + 4z \\ \hline x = 1, y = 2, z = 3 & 51 & 51 & 19 \\ \hline x = 10, y = 20, z = 30 & 510 & 510 & 190 \\ \hline x = 23, y = 60, z = 10 & 549 & 549 & 229 \\ \hline x = 14, y = 0, z = 1 & 54 & 54 & 46 \\ \hline x = 5, y = 9, z = 32 & 453 & 453 & 161 \\ \hline \end{array} \][/tex]

### Step 3: Identify Equivalent Expressions

The expressions [tex]$ 3(x + 2y + 4z) $[/tex] and [tex]$ 3x + 6y + 12z $[/tex] are equivalent, as shown by the calculations.

So the equivalent expressions are:
[tex]\[ 3(x + 2y + 4z) \text{ and } 3x + 6y + 12z \][/tex]