4.1 Algebraic expressions

$9a^2-6a-12$ contains three terms. Some of the factors in each term are

$\begin{array}{ll}\text{first}\text{term:}\hfill & 9\text{and}{a}^2,\text{or},9\text{and}a\text{and}a\hfill \\ \text{second}\text{term:}\hfill & -6\text{and}a\hfill \\ \text{third}\text{term:}\hfill & -12\text{and}1,\text{or},12\text{and}-1\hfill \end{array}$

$14x^{5}y+{(a+3)}^2$ contains two terms. Some of the factors of these terms are

$\begin{array}{ll}\text{first}\text{term:}\hfill & 14,x^5,y\hfill \\ \text{second}\text{term:}\hfill & (a+3)\text{and}(a+3)\hfill \end{array}$

$5x^3-7x^3+14x^3$ .

The factor $x^3$ appears in each and every term. The expression $x^3$ is a common factor.

$4x^2+7x$ .

The factor $x$ appears in each term. The term $4x^2$ is actually $4xx$ . Thus, $x$ is a common factor.

$12x{y}^2-9xy+15$ .

The only factor common to all three terms is the number 3. (Notice that $12=3\cdot 4,9=3\cdot 3,15=3\cdot 5$ .)

$3(x+5)-8(x+5)$ .

The factor $(x+5)$ appears in each term. So, $(x+5)$ is a common factor.

$45x^3{(x-7)}^2+15x^2(x-7)-20x^2{(x-7)}^5$ .

The number 5, the $x^2$ , and the $(x-7)$ appear in each term. Also, $5x^2(x-7)$ is a factor (since each of the individual quantities is joined by a multiplication sign). Thus, $5x^2(x-7)$ is a common factor.

$10x^2+9x-4$ .

There is no factor that appears in each and every term. Hence, there are no common factors in this expression.

$12x$ means there are $12x\text{'}\text{s}$ .

$4ab$ means there are four $ab\text{'}\text{s}$ .

$10(x-3)$ means there are ten $(x-3)\text{'}\text{s}$ .

$1y$ means there is one $y$ . We usually write just $y$ rather than $1y$ since it is clear just by looking that there is only one $y$ .

$7a^3$ means there are seven $a^{3\text{'}}\text{s}$ .

$5ax$ means there are five $ax\text{'}\text{s}$ . It could also mean there are $5ax\text{'}\text{s}$ . This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression $5ax$ we must ask ourselves "Are we working with the quantity $ax$ or the quantity $x$ ?".

$6x^2{y}^9$ means there are six $x^2{y}^{9\text{'}}\text{s}$ . It could also mean there are $6x^2{y}^{9\text{'}}\text{s}$ . It could even mean there are $6y^9{x}^{2\text{'}}\text{s}$ .