Integral Compound Expressions.

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

84. Parentheses or Brackets.

We have for positive numbers (§§ 37, 38): {2} a + (b + c) &= a + b + c,\qquad & \therefore a + b + c &= a + (b + c); \ a + (b - c) &= a + b - c, & \therefore a + b - c &= a + (b - c); \ a - (b + c) &= a - b - c, & \therefore a - b - c &= a - (b + c); \ a - (b - c) &= a - b + c, & \therefore a - b + c &= a - (b - c).

That is, a parenthesis preceded by $+$ may be removed without changing the sign of any term within the parenthesis; and any number of terms may be enclosed within a parenthesis preceded by the sign $+$, without changing the sign of any term.

A parenthesis preceded by the sign $-$ may be removed, provided the sign of every term within the parenthesis is changed, namely, $+$ to $-$, and $-$ to $+$; and any number of terms may be enclosed within a parenthesis preceded by the sign $-$, provided the sign of every term enclosed is changed.

The same laws hold for negative numbers.

Expressions may occur having a parenthesis within a parenthesis. In such cases parentheses of different shapes are used, and the beginner when he meets with a branch of a parenthesis $($, or bracket $[$, or brace $\{$, must look carefully for the other part, whatever may intervene; and all that is included between the two parts of each parenthesis must be treated as the sign before it directs, without regard to other parentheses. It is best to remove each parenthesis in succession, beginning with the innermost.

Exercise 20.

Remove the brackets and collect the like terms:

1. $a-b-(b-c)-a+2b$.

2. $x-[x-(a-b)+a-y]$.

3. $3x-\backslash \{2y-[-7c-2x]+y\backslash \}$.

4. $5a-[7-(2b+5)-2a]$.

5. $x-[2x+(3a-2x)-5a]$.

6. $x-[15y-(13z+12x)]$.

7. $2a-b+[4c-(b+2c)]$.

8. $5a-\backslash \{b+[3c-(2b-c)]\backslash \}$.

9. $7x-\backslash \{5y-[3z-(3x+z)]\backslash \}$.

10. $(a-b+c)-(b-a-c)+(a+b-2c)$.

11. $3x-[-2y-(2y-3x)+z]+[x-(y-2z-x)]$.