4.4 Combining polynomials using addition and subtraction

$6\text{houses}+4\text{houses}=10\text{houses}$ . 6 and 4 of the same type give 10 of that type.

$6\text{houses}+4\text{houses}+2\text{motels}=10\text{houses}+2\text{motels}$ . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type.

Suppose we let the letter $x$ represent "house." Then, $6x+4x=10x$ . 6 and 4 of the same type give 10 of that type.

Suppose we let $x$ represent "house" and $y$ represent "motel."

$2x+5x+3x$ .
There are $2x\text{'}\text{s}$ , then 5 more, then 3 more. This makes a total of $10x\text{'}\text{s}$ .

$2x+5x+3x=10x$

$7x+8y-3x$ .
From $7x\text{'}\text{s}$ , we lose $3x\text{'}\text{s}$ . This makes $4x\text{'}\text{s}$ . The $8y\text{'}\text{s}$ represent a quantity different from the $x\text{'}\text{s}$ and therefore will not combine with them.

$7x+8y-3x=4x+8y$

$4a^3-2a^2+8a^3+a^2-2a^3$ .
$4a^3,8a^3,$ and $-2a^3$ represent quantities of the same type.

$4a^3+8a^3-2a^3=10a^3$

$-2a^2$ and $a^2$ represent quantities of the same type.

$-2a^2+a^2=-a^2$

Thus,

$4a^3-2a^2+8a^3+a^2-2a^3=10a^3-a^2$