Integral Compound Expressions.

$& 2 a^{3} - 3 a^{2} + 2 a - 1 - a^{3} - 2 a^{2} - 3 a + 5 & = 2 a^{3} - a^{3} - 3 a^{2} - 2 a^{2} + 2 a - 3 a - 1 + 5 & = a^{3} - 5 a^{2} - a + 4 .$

This process is more easily performed by writing the subtrahend below the minuend, mentally changing the sign of each term in the subtrahend, and adding. $r * 3 c r 2 a^{3} & - & 3 a^{2} & + & 2 a & - & 1 a^{3} & + & 2 a^{2} & + & 3 a & - & 5 \hline a^{3} & - & 5 a^{2} & - & a & + & 4$

By changing the sign of each term in the subtrahend, the coefficient of $a^{3}$ will be $2 - 1$ , or $1$ ; the coefficient of $a^{2}$ will be $- 3 - 2$ , or $- 5$ ; the coefficient of $a$ will be $2 - 3$ , or $- 1$ ; the last term will be $- 1 + 5$ , or $4$ .

Again, suppose it is required to subtract $x^{5} - 2 a x^{4} - 3 a^{2} x^{3} + 4 a^{3} x^{2}$ from $4 a^{3} x^{2} - 2 a^{2} x^{3} - 5 a x^{4}$ . Here terms which are alike can be written in columns, as before: $r * 3 c r & - & 5 a x^{4} & - & 2 a^{2} x^{3} & + & 4 a^{3} x^{2} x^{5} & - & 2 a x^{4} & - & 3 a^{2} x^{3} & + & 4 a^{3} x^{2} \hline - x^{5} & - & 3 a x^{4} & + & a^{2} x^{3} & &$

There is no term of $x^{5}$ in the minuend, hence the coefficient of $x^{5}$ in the result will be $0 - 1$ , or $- 1$ ; the coefficient of $a x^{4}$ will be $- 5 + 2$ , or $- 3$ ; the coefficient of $a^{2} x^{3}$ will be $- 2 + 3$ , or $+ 1$ ; the coefficient of $a^{3} x^{2}$ will be $- 4 + 4$ , or $0$ , and therefore the term $a^{3} x^{2}$ will not appear in the result.

Exercise 19.

Subtract:

$a - 2 b + 3 c$ from $2 a - 3 b + 4 c$ .

$a - 3 b - 5 c$ from $3 a - 5 b + c$ .

$2 x - 4 y + 6 z$ from $4 x - y - 2 z$ .

$5 x - 11 y - 3 z$ from $6 x - 7 y + 2 z$ .

$a b - a c - b c + b d$ from $a b + a c + b c + b d$ .

$3 a b + 2 a c - 3 b c + b d$ from $5 a b - a c + b c + b d$ .

$2 x^{3} - x^{2} - 5 x + 3$ from $3 x^{3} + 2 x^{2} - 3 x - 5$ .

$7 x^{2} - 5 x + 1 - a$ from $x^{3} - x + 1 - a$ .

$7 b^{3} + 8 c^{3} - 15 a b c$ from $9 b^{3} + 3 a b c - 7 c^{3}$ .

$x^{4} + x - 5 x^{3} + 5$ from $7 - 2 x^{2} - 3 x^{3} + x^{4}$ .

$a^{3} + b^{3} + c^{3} - 3 a b c$ from $3 a b c + a^{3} - 2 b^{3} - 3 c^{3}$ .

$2 x^{4} - 5 x^{2} + 7 x - 3$ from $x^{4} + 2 - 2 x^{3} - x^{2}$ .

$1 - x^{5} - x + x^{4} - x^{3}$ from $x^{4} + 1 + x + x^{2}$ .

$a^{3} - b^{3} + 3 a^{2} b - 3 a b^{2}$ from $a^{3} + b^{3} - a^{2} b - a b^{2}$ .

$a^{2} b - a b^{2} - 3 a^{3} b^{3} - b^{4}$ from $b^{4} - 5 a^{3} b^{3} - 2 a b^{2} + a^{2} b$ .