Integral Compound Expressions.

If an algebraic expression contains only integral forms, that is, contains no letter in the denominator of any of its terms, it is called an integral expression.

Thus, $x^3+7c{x}^2-c^3-5c^{2}x$, is an integral expression.

Integral and fractional expressions are so named on account of the form of the expressions, and with no reference whatever to the numerical value of the expressions when definite numbers are put in place of the letters.

81. Addition of Integral Compound Expressions.

The addition of two algebraic expressions can be represented by connecting the second expression with the first by the sign $+$. If there are no like terms in the two expressions, the operation is algebraically complete when the two expressions are thus connected.

If, for example, it is required to add $m+n-p$ to $a+b+c$, the result will be $a+b+c+(m+n-p)$; or, removing the parenthesis (§ 37), $a+b+c+m+n-p$.

If there are like terms in the expressions, the like terms can be collected; that is, every set of like terms can be replaced by a single term with a coefficient equal to the algebraic sum of the coefficients of the like terms.

1. Add $6x^2+5x+4$ to $x^2-4x-5$.

This process is more conveniently represented by arranging the terms in columns, so that like terms shall stand in the same column, as follows: $$\begin{gathered} r*2cr{x}^2\&-\&4x\&-\&5 \\ 6x^2\&+\&5x\&+\&4 \\ \mathrm{\backslash hline}7x^2\&+\&x\&-\&1 \end{gathered}$$

The coefficient of $x^2$ in the result will be $6+1$, or $7$; the coefficient of $x$ will be $-4+5$, or $1$; and the last term is $-5+4$, or $-1$.

When the coefficient of a term is $1$, it is not written, but understood.

2. Add $2c^3-5c^{2}d+6c{d}^2+d^3$; $c^3+6c^{2}d-5c{d}^2-2d^3$; and $3c^3-c^{2}d-7c{d}^2-3d^3$. $$\begin{gathered} r*3cr2c^3\&-\&5c^{2}d\&+\&6c{d}^2\&+\&d^3 \\ {c}^3\&+\&6c^{2}d\&-\&5c{d}^2\&-\&2d^3 \\ 3c^3\&-\&c^{2}d\&-\&7c{d}^2\&-\&3d^3 \\ \mathrm{\backslash hline}6c^3\&\&\&-\&6c{d}^2\&-\&4d^3 \end{gathered}$$

The coefficient of $c^3$ in the result will be $2+1+3$, or $6$; the coefficient of $c^{2}d$ will be $-5+6-1$, or $0$; therefore $c^{2}d$ will not appear in the result; the coefficient of $c{d}^2$ will be $6-5-7$, or $-6$; and the coefficient of $d^3$ will be $1-2-3$, or $-4$.

Exercise 18.

Find the sum of:

1. $a^2-ab+b^2$; $a^2+ab+b^2$.