By Harley Flanders and Justin J. Price (Auth.)

ISBN-10: 012259665X

ISBN-13: 9780122596650

Algebra and Trigonometry

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BASIC ALGEBRA (b) Try (ax + b)(cx + d). By formula (2), we need ac = 2. So try a = 1 and c = 2, that is, (x + è)(2x + d). Now we need W = 3 and 2b + d = - 7 . A little trial and error shows that b = — 3 and d — — 1 work, so 2*2 _ 7jc + 3 = (x _ 2)(2χ - 1). Answer (a) (x + \)(x + 9) (b) (x - 3)(2JC - 1). 2 Factor x5 — 8JC3 + 16*. SOLUTION First remove the common factor x: x5 - 8x 3 + \6x = x(x4 - 8x 2 + 16). The polynomial in parentheses is a perfect square, (x2 — 4) 2 . Also, by formula (3), we have x2 — 4 = (x + 2)(x — 2).

We obtain 24xy, not &xy; therefore the given expression (d) is not a square. Answer (a) (x - 5)2 (c) (6x + ^j) 2 (b) (xy + 3z2)2 (d) not a square. 37 38 1. BASIC ALGEBRA EXERCISES Compute: 1. (x + 2j + 3) + ( 4 x - 7 - 1) 3. ( - 4 x + 5^ + 6z) - (x - y + z + 2) 5. (x2 + 3>;2) + ( x 2 + 7 2 - 3 y ) 6. (x 2 + 6XJ + 2 ^ 2 ) - (x 2 + 4xy-y* - x 2. ( 2 x - j + 8) + (3x + 2 ; ; - 5 ) 4. (7 + 2z) - (3* + y + z + 1) -2) 7. (x2 + y2 + z2 + xy + j z + xz) - (3x2 + j 2 + z 2 + xj + x + y) 8. x(3x2 + xy3 + 7z 4 ) — j>(xz4 + xy + x2y2).

Sometimes identities are called formulas. But usually the word formula is used for an equation in which the letters involved denote physical or geometrical 49 50 2. EQUATIONS AND INEQUALITIES quantities. For instance, d—rt (distance) = (rate)(time) A = \bh (area) = ^(base)(height) E = mc2 (energy) = (mass)(speed of light)2 1 + 2 + 3 + · · · + n = \n(n + 1) (sum of first n positive integers) = \n{n + 1) are formulas expressing laws of nature when the letters have the indicated meanings. Solution of Equations The easiest way to solve an equation is by inspection—if the equation is simple enough.