Page:Elementary algebra (1896).djvu/70

52 Rh The expression = {aj^- (W) + (hx^- dx"^) + (bx -ex -2 x) + (7 - c) = x3(« - d) + x^{b - d) + x{h - c - 2) + (7 - c) = (a - cZ)x3 + (6 - d)x2 + (5 _ c - 2)x + 7 - c.

In this last result the compound expressions a — d, b —d, b — c — 2 are regarded as the coefficients of x^3, x^2, and x respectively.

Ex. 2. In the expression - a^2x - 7 a + a^2y +3 - 2x - ab bracket together the powers of a so as to have the sign — before each bracket.

The expression = - (a^2x - a^2y) - (7a + ab) - (2 x - 3) = -a^2(ax-y)-a(7 + b))-(2x-3) = -(x-y)a^2-( 7 + b)a-(2x-3).

EXAMPLES VI. c.

In the following expressions bracket the powers of x so that the signs before all the brackets shall be positive :

1. ax^ + 6x^2 + 5 + 2 bx - 5 x^2 + 2 x - 3 x. 2. 3 bx2 - 7 - 2 x + ab + 5 ax^ + cx - 4 x^2 - bx^. 3. 2 - 7 x+3 + 5 ax^2 - 2 cx + 9 ax^3 + 7 x - 3 x^2.

In the following expressions bracket the powers of x so that the signs before all the brackets shall be negative :

4. ax^2 + 5 x^3 - a^2x^4 - 2 bx^3 - 3x^2 - bx^4. 5. 7 x^3 - 3 c^2x - abx^ + 5 ax + 7 x^ - abcx^. 6. 3 b^2x^ - bx - ax^ - cx^ - 5 c^x - 7 x^.

Simplify the following expressions, and in each result regroup the terms according to powers of x :

7. ax^3 - 2 cx - [bx^2 - {cx - dx - (bx^3 -cx ^2)}- (cx^2 - bx)]. 8. 5 ax^3 - 7(bx - cx^2) - {6 bx^2 - (3 ax^2 + 2 ax) -4 cx^3}. 9. ax^2 - 3 {- ax3 + 3 bx - 4 [cx^3 - (ax - bx^2)]}. 10. x^5 - 4 bx^ - [12 ax - 4 { 3 bx^ - 9(cx-bx^)- ax^}].

71. In certain cases of addition, multiplication, etc., of expressions which involve literal coefficients, the results may be more conveniently written by grouping the terms according to powers of some common letter.

Ex. 1. Add together ax^3 — 2 bx^2+3, bx— cx^3-x^2, and x^3 — ax^2+cx.

The sum = ax^3 — 2 bx^2 + 3+ bx — cx^3 — x^2 + x^3 — ax^3 + cx = ax^3 - cx^3 + x^3 - ax^2 - 2 bx^2 - x^2 + bx + cx +3. =(a-c +1)x^3 (a + 2 b +1)x^2 + (b + c)x + 3.