Algebraic Equations 9781423223085, 9781423222668

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WORLD’S #1 ACADEMIC OUTLINE

FORMULAS, PROPERTIES & OPERATIONS

PROPERTIES OF REAL NUMBERS

PROPERTIES OF EQUALITY

For any real numbers a, b, and c: 1. Closure a. For addition: a + b is a real number b. For multiplication: a • b is a real number 2. Commutative Property a. For addition: a+b=b+a b. For multiplication: ab = ba 3. Associative Property a. For addition: a + (b + c) = (a + b) + c b. For multiplication: a(bc) = (ab)c 4. Identity Property a. For addition: a + 0 = a and 0 + a = a b. For multiplication: a • 1 = a and 1 • a = a 5. Inverse Property a. For addition: a + (–a) = 0 and (–a) + a = 0 b. For multiplication: a • 1a = 1 and 1a • a = 1 6. Distributive Property a(b + c) = ab + ac; a(b – c) = ab – ac; and ab + ac = a(b + c); ab – ac = a(b – c) 7. Multiplication Property of Zero 0 • a = 0 and a • 0 = 0 8. Double Negative Property –(–a) = a or –1(–1 • a) = a Note: While a + b = b + a is true and correct, a – b = b – a is not true and not correct. Likewise, a • b = b • a is correct and a ÷ b = b ÷ a is not correct.

For any real numbers a, b, and c: 1. Reflexive Property: a = a 2. Symmetric Property: If a = b, then b = a. 3. Transitive Property: If a = b and b = c, then a = c. 4. Addition Property: If a = b, then a + c = b + c for any value of c. 5. Multiplication Property: If a = b, then ac = bc for any value of c. 6. Proportion Property: If a = c , then ad = bc; a = b ; b = d ; c = d ; a ± b = c ± d ; b d c d a c a b b d a−b = c −d . a+b c+d

OPERATIONS OF REAL NUMBERS 1. Absolute value a. |x| = x if x ≥ 0 and –x if x < 0. It is always a positive numerical value. b. |x| = |–x| c. |x| ≥ 0 d. |x – y| = |y – x| 2. Addition: If the signs of the numbers are the same, add. If the signs of the numbers are different, subtract. In both cases, the answer has the sign of the number with the larger absolute value. 3. Subtraction: Change subtraction to addition of the opposite number, then follow the addition rules. 4. Multiplication: Multiply the numbers, then determine the sign of the answer. Remember: negative • negative = positive; positive • positive = positive; negative • positive = negative; positive • negative = negative; if the signs are the same, the answer is positive; if signs are different, the answer is negative. 5. Division: Divide the numbers, then determine the sign of the answer using the same sign rules that apply to multiplication.

OPERATIONS OF COMPLEX NUMBERS 1. Definition: a + bi where a, b ∈ real numbers (can be negatives) and i = −1 . Note: i2 = –1 and is a real number, not an imaginary number. 2. Addition: (a + bi) + (c + di) = (a + c) + (b + d)i 3. Subtraction: (a + bi) – (c + di) = (a – c) + (b – d)i 4. Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i Note: (a ± bi) 2 = (a ± bi)(a ± bi) = a2 ± abi ± abi + b2 i2 = (a2 – b2) ± 2abi and not the common mistake (a ± bi) 2 = a2 ± b2 i2 . ac + bd ) + (bc − ad ) i 5. Division: a + bi = a + bi • c − di = ( c + di c + di c − di c2 + d 2

PROPERTIES OF INEQUALITY For any real numbers a, b, and c: 1. Trichotomy Property: Either a = b, a < b, or a > b. 2. Transitive Property: If a < b and b < c, then a < c; also, if a > b and b > c, then a > c. 3. Addition Property: If a < b, then a + c < b + c; also, if a > b, then a + c > b + c for any value of c. 4. Multiplication Property: If c > 0 and a < b, then ac < bc. If c > 0 and a > b, then ac > bc. If c < 0 and a < b, then ac > bc. If c < 0 and a > b, then ac < bc. If c = 0, then ac = bc = 0.

OPERATIONS OF ALGEBRAIC EXPRESSIONS Like or similar terms are terms with the same variables having the same exponent values. ADDITION & SUBTRACTION OF POLYNOMIALS ax + bx = (a + b)x or ax – bx = (a – b)x; if the variables and exponents are the same, add or subtract the numbers in front (coefficients) without changing the variable. ADDITION & SUBTRACTION OF RATIONAL EXPRESSIONS (FRACTIONS) b 1. ax ± bx = a ± x ; if the denominators are the same, add the numerators only. a b ay bx ay ± bx 2. x ± y = xy ± yx = xy ; if the denominators are not the same, multiply each fraction by one in the required form, then add the numerators. If the denominators are polynomials, then factoring them first will help determine the least common denominator.

Note: Avoid these common mistakes:

;

;

; and

.



Instead, the correct methods are these:  ; ; 1 1 1 x x 1 x 1 y 1 1 . x = • = ; and = ÷ = • = 2 2 1 2 y x 1 x y xy MULTIPLICATION RULES 1. x • x • x • … • x = xn when the number of x variables = n. 2. x0 = 1 when x ≠ 0. 3. x1 = x 4. xm • xn = xm+n; also axm • bxn = abxm+n; multiply coefficients and variables. 5. (xm) n = xmn; also (xmy p) n = xmny pn; powers of powers of monomials can be done by multiplying exponents. 6.  x   y

m

m = xm y

7. If au = av, then u = v. 8. If au = bu for a ≠ 0, then a = b. Note: Avoid these common mistakes: (x2)3 = x5 and (3x)2 = 3x2. Instead, the correct methods are these: (x2)3 = x2•3 = x6 and (3x)2 = 32 x2 = 9x2 , but (x + 3)2 = (x + 3)(x + 3) = x2 + 6x + 9. DISTRIBUTIONS 1. Type 1: a(x + y) = ax + ay; a(x – y) = ax – ay Note: Avoid these common mistakes: x – (y + z) = x – y + z and x – (y – z) = x – y – z. Instead, the correct methods are these: x – (y + z) = x – y – z and x – (y – z) = x – y + z. 2. Type 2: (a + b)(x + y) = a(x + y) + b(x + y) = ax + ay + bx + by a. This type, 2 terms times 2 terms, can also be done using the FOIL rule. b. FOIL: First term times first term, Outer term times outer term, Inner term times inner term, Last term times last term, (a + b)(x + y). Note: Avoid this mistake: (x + y) 2 = x2 + y2 . Instead, the correct method is this: (x + y) 2 = (x + y)(x + y) = x2 + xy + xy + y2 = x2 + 2xy + y2 , but (xy) 2 = xy • xy = x2y2 is correct because it is only one term, not two. 1

OPERATIONS OF ALGEBRAIC EXPRESSIONS (continued)

3. Type 3: (a + b)(x + y + z) = a(x + y + z) + b(x + y + z) = ax + ay + az + bx + by + bz 4. The Binomial Theorem: The expansion of (x + y) n, where n is a counting number, is a1 xn + a2 xn–1y + a3 xn–2y2 +…+ an–1 xyn–1 + anyn where a1, a2 , a3,…, an are found in Pascal’s triangle: 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 and so on… n (n − 1) n–2 2 5. The Binomial Formula: (x + y) n = x n + n x n–1y + x y +…+ 1 1• 2 n (n − 1) (n − 2)…(n − r + 1) n–r r n x y +…+ y 1 • 2 • 3… r  n n! 6. The Binomial Coefficient:  r  = r ! n − r ! ( ) 7. Other special types of distributions are listed under the Factors section. RATIONAL EXPRESSIONS a • c = ac b d bd polynomial polynomial 2. must have all of the numerators and denominators • polynomial polynomial factored so identical factors (one in a numerator and the other in a denominator) can be cancelled before multiplying. 1.

DIVISION x± y = 1; whenever the numerator and denominator are the same and not 1. xx = 1 and x± y zero, the fraction equals one. 2. ax = xy ; common factors cancel to equal one. ay m m 3. x n = x m − n if m > n and x n = n1− m if m < n. x x x 4. a ÷ c = a • d = ad ; if any of the numerators or denominators are polynomials, like b d b c bc ax 2 ± bx ± c , they must first be factored so identical factors can be cancelled as dx 2 ± ex ± f indicated in the Multiplication Rules section (p. 1). a 5. bc = a ÷ c = a • d ; complex fractions are indicated by division. b d b c d a±c ad ± cb ad ± cb b d = bd bd = bd = ad ± cb ÷ eh ± gf = ad ± cb • fh = fh (ad ± cb) 6. eh ± gf bd fh bd eh ± gf bd (eh ± gf ) e ±g eh ± gf fh f h fh fh

−1

⎛1⎞ 1 and ⎜⎜⎜ ⎟⎟⎟ = − . Instead, the 2 ⎝ 2 ⎟⎠ −1

FRACTION EXPONENTS/RADICAL EXPRESSIONS 1. a

= n am and

n

1

n

m

am = a n .

2.

a = a n = b where bn = a; if n is even and a < 0, even and a ≥ 0, n a is the nonnegative number b.

3.

n

a n = a ; (a n m

n

1 n

n

1

n n n 4. (ax) n = n (ax) = am x m = am • x m ; and m

m

m  1 m n 5. (ax) n = (ax) n  = (ax) ; and m

n

n

m  7.  =n xp =  y p  y

n

= a n = a1 = a . m

m

m

am • n x m = n am x m = (ax) n = a n x n . 1 m

m

(ax)m = (ax) n  = (ax) n .

6. (ax ± by) n = n (ax ± by)m , BUT it does NOT equal 1 n xm 

a is not a real number, and if n is

) n = a n = a1 = a if n is odd and |a| if n is even, and (a n )

n

(ax)m

m ± n (by) .

m

n

x m = x n , BUT if the expression is to be written in simplest form, p n yp yn a radical expression cannot be left in the denominator. 1

n xm = n xm =  xm   p  p n yp y y

8.

n

9.

nm

y can be

12. cannot be combined unless the radicals n x and simplified to be the same, making the indices n and m equal.

m

x can be

Note: Avoid these common mistakes:

and

Instead, the correct methods are these:

 . , and

cannot be simplified. FACTORS 1. Linear Factor Theorem: A polynomial of degree n ≥ 1 can be written as the product of n linear factors, P(x) = a(x – r1)(x – r2)…(x – rn). 2. Greatest common factor: ax ± ay = a(x ± y) 3. Quadratic trinomials: a. x2 ± bx ± c = (x + h)(x + k) where h • k = c and h + k = b; h and k can be either positive or negative numbers. b. ax2 ± bx ± c = (mx + h)(nx + k) where a ≠ 1 and m • n = a, h • k = c, m • k + h • n = b; m, n, h, and k can be positive or negative numbers. 4. Perfect square: x2 ± 2xy + y2 = (x ± y)(x ± y) = (x ± y) 2 5. Perfect cube: x3 ± 3x2y + 3xy2 ± y3 = (x ± y) 3 6. Difference of two squares: x2 – y2 = x2 + xy – xy – y2 = (x + y)(x – y) 7. Sum of two squares: x2 + y2 = (x + yi)(x – yi), where i = −1 and is an imaginary number. 8. Difference of two cubes: x3 – y3 = (x – y)(x2 + xy + y2) 9. Sum of two cubes: x3 + y3 = (x + y)(x2 – xy + y2) 10. Grouping a. 2-2 grouping: ax ± ay ± bx ± by = a(x ± y) ± b(x ± y) = (a ± b)(x ± y) b. 3-1 grouping: x2 ± 2cx + c2 – y2= (x ± c) 2 – y2 = (x ± c + y)(x ± c – y) c. 1-3 grouping: y2 – x2 ± 2cx – c2 = y2 – (x ± c) 2 = (y + x ± c)(y – x ± c) 11. Partial-fraction decomposition rules a. Linear factors: For each distinct factor of the form (ax + b) m in the denominator Q(x), introduce the sum of m partial fractions. A1 A2 Am + where A1, A2 ,…, Am are constants. ax + b (ax + b)2 +…+ ax ( + b)m b. Quadratic factors: For each distinct factor of the form (ax2 + bx + c) m in the denominator Q(x), introduce the sum of m partial fractions A2 x + B2 Am x + Bm A1 x + B1 + +…+ where A1, 2 2 2 ax + bx + c (ax + bx + c (ax2 + bx + c m A2 ,…, Am and B1, B2 ,…, Bm are constants.

)

SOLVING EQUATIONS

⎛1⎞ 2 1 2 1−1 2 correct methods are these: 2x −1 = 21 • x −1 = • = and ⎜⎜ ⎟⎟⎟ = −1 = = 2 , but ⎜ 1 x x 1 2 ⎝ 2 ⎟⎠ is correct. m n

n

)

NEGATIVE EXPONENTS = xm.

1. x–m = 1m and −1m x x 1 1 2. (x ± y) –m = and = (x ± y) m. ( x ± y)m ( x ± y)− m Note: Avoid these common mistakes:

11. cannot be combined unless the radicals n x and simplified to be the same, making the radicands x and y equal.

a = nm a

10. 2

STEPS TO SOLVE FIRST-DEGREE WITH ONE VARIABLE 1. Simplify the left side of the equals sign. 2. Simplify the right side of the equals sign. 3. Apply inverse operations until the variable is isolated on either side of equals sign. Note: If the statement is an inequality and multiplication or division by a negative number was used to distribute throughout the entire inequality, then the inequality symbol must be reversed to keep the statement true and the solution correct. 4. For absolute value equations, |m| = a where m is an algebraic expression in terms of x and a is a nonnegative number (if a is a negative number then there is no solution), then the one equation |m| = a becomes two equations (m) = a and –(m) = a; solve both equations; the solution is the union of the solutions (i.e., both solutions should solve the equation |m| = a). 5. For absolute value inequalities of the form |m| > a where m is an algebraic expression in terms of x and a is a real number, then the one inequality becomes (m) > a ∪ –(m) > a; then solve the inequalities; this method is the same for |m| ≥ a . 6. For absolute value inequalities of the form |m| < a where m is an algebraic expression in terms of x and a is a nonnegative number (if a is a negative number then there is no solution), then the one inequality becomes (m)  0 and a ≠ 1; when x = 0 the y-intercept is 1; graphs are continuous curves that increase if a > 1 and decrease if a  0 and a ≠ 1; when x = 0, there is no y-intercept; graphs are continuous curves that increase if a > 1 and decrease if a  0, a ≠ 1, and n is an expression in terms of x. 2. Inverses of exponential functions are logarithmic functions. LOGARITHMIC FUNCTIONS 1. For all positive numbers a, where a ≠ 1, y = loga x if and only if x = a y. 2. The common logarithm, log x, has a base of 10, so a = 10 in the definition of log. 3. The natural logarithm, ln x, has a base equal to the number e; a = e ≈ 2.71828, in the definition of log. 4. Properties with a > 0 and a ≠ 1: a. aloga x = x b. logaa x = x c. logaa = 1 d. loga1 = 0 e. If logau = logav, then u = v. f. If logau = logbu and u ≠ 1, then a = b. g. logaxy = logax + loga y  x h. loga  y  = logax – loga y

CONIC SECTIONS The general form of the equation of a conic section with axes parallel to the coordinate axes is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A and C are not both zero. 1. Lines y −y y −y ∆y rise a. Slope = m = x2 − x1 = x1 − x2 = = ∆x run 2 1 1 2 b. Point-slope form of a linear equation: y – y1 = m(x – x1). c. Slope-intercept form of a linear equation: y = mx + b, where m is the slope of the line and b is the y-intercept (point where the line crosses the y-axis). d. Standard form of a linear equation: ax + by = c, where a, b, and c are integers. e. Vertical lines have the equation x = a, where a is the x-value where the line crosses the x-axis. f. Horizontal lines have the equation y = a, where a is the y-value where the line crosses the y-axis. g. Two lines are parallel if and only if their slopes are equal; m1 = m2 . h. Two lines are perpendicular if and only if their slopes are negative 1 1 reciprocals, and m2 = − m and m1 = − m . 2 1 i. If m > 0, the line is increasing. j. If m < 0, the line is decreasing. k. If m = 0, the line is horizontal. l. If m is undefined, the line is vertical. 2. Parabolas a. General equation: y = a(x – h) 2 + k (opens up-down), or x = a(y – k) 2 + h (opens left-right) b. Standard form: (x – h) 2 = 4p(y – k) (opens up-down), or (y – k) 2 = 4p(x – h) (opens left-right) 1 c. Where (h, k) is the vertex, (h, k ± p) or (h ± p, k) is the focus with p = 4 a , y = k ± p or x = h ± p is the directrix, and x = h or y = k is the line of symmetry. 3. Circles a. General equation: (x – h) 2 + (y – k) 2 = r2 b. (h, k) is the center of the circle and r is the radius. 4. Ellipses ( x − h)2 ( y − k )2 a. General equation: + =1 a2 b2

()

i. loga 1 = –logax x j. logaxn = n(logax), where n is a real number. k. Change of base rule: If a > 0, a ≠ 1, b > 0, b ≠ 1, and x > 0, then (loga x) . logbx = (loga b) l. ln x =

, and have a line of symmetry of

(log x) (log e)

GRAPHING FUNCTIONS 1. Intercepts a. Make x equal zero and solve for f(x) to find the y- or f(x)-intercept. b. Make y equal zero and solve for x to find the x-intercept. 2. Symmetry a. A function is symmetric to the y-axis if points (x, y) and (–x, y) are both on the graph and f(a) = f(–a) for all values of a; thus, the function is an even function. b. A function is symmetric to the origin if points (x, y) and (–x, –y) are both on the graph and f(–a) = –f(a) for all values of a; thus, the function is an odd function. c. A function is symmetric to the x-axis if points (x, y) and (x, –y) are both on the graph; however, it is a relation and not a function unless y = 0 or f(x) = 0. 3. Special functions a. Linear functions in slope-intercept form are f(x) = mx + b, where m is the slope and b is the y-intercept. 4

COORDINATE PLANE (continued)

b. Where (h, k) is the center, a is the horizontal movement left and right from the center to points on the ellipse, and b is the vertical movement up and down from the center to points on the ellipse. c. Additionally, when a > b, then major axis is horizontal and foci are (h ± c, k); where c2 = a2 – b2 and where a 1 or |r| = 1, the sum does 1− r not exist. 10. The r th term of the binomial expansion of (x + y) n is r −1 n! x n −(r −1) y . [n − (r − 1)]!(r − 1)!

WORK 1. W is the ratio of time to complete the job together by a team of people or machines compared to the time to complete the job alone by one person or one machine. 2. One, 1, represents the whole job. 3. Formula: W1 + W2 +…+ Wn = 1 DISTANCE 1. d is distance. 2. r is rate or speed. 3. t is time as indicated in the rate; for example, miles per hour or meters per second. 4. Formula: d = rt 5. These relationships can be used depending on the situation described in the problem: a. dto = dreturning b. d1 + d2 = dtotal PROPORTIONS & VARIATIONS 1. a, b, c, and d are quantities specified in the problem. 2. k ≠ 0 3. Formulas: a. Proportion: a = c ; cross-multiply to get ad = cb. b d b. Direct variation: y = kx c. Inverse variation: y = kx d. Combined variation: y = kx z ; y varies directly as x and inversely as z.

2. ∑ cak = c ∑ ak , where c is a constant. k =1 n

k =1

k =1

)

5

TRIGONOMETRY

TRIG IDENTITIES & FORMULAS 1. Reciprocals

TRIG WITH TRIANGLES 1. Special right triangle side ratios Note: The longest side of a right triangle is always the hypotenuse; the other two sides are the legs.

a. b.

a. 30° – 60°– 90° triangles have side lengths with the ratios of 1 : 3 : 2 .

c.

b. 45° – 45°– 90° triangles have side lengths with the ratios of 1 :1 : 2 . 2. Trig function definitions using right triangles

2. Cofunctions

a. Sine of an angle A:

a.

b. Cosine of an angle A:

b.

c. Tangent of an angle A:

c.

d. Cosecant of an angle A:

3. Basic identities a.

e. Secant of an angle A:

b. sin 2 A + cos2 A = 1 or sin 2 A = 1 – cos2 A or cos2 A = 1 – sin 2 A c. tan 2 A + 1 = sec2 A or sec2 A – tan 2 A = 1 or tan 2 A = sec2 A – 1 d. cot2 A + 1 = csc2 A or csc2 A – cot2 A = 1 or cot2 A = csc2 A – 1 4. Addition and subtraction formulas a. sin(A ± B) = sinA cosB ± cosA sinB b. cos(A ± B) = cosA cosB  sinA sinB c. tan A ± B = tan A ± tan B 1  tan A tan B 5. Negatives a. sin(–A) = –sinA

f. Cotangent of an angle A: 3. Solving nonright triangles (acute and oblique) a. The Law of Sines:

; use any two of these ratios in an

(

equation, where a is the side opposite angle A, b is the side opposite angle B, and c is the side opposite angle C. b. The Law of Cosines: Use any one of the following, where a is the side opposite angle A, b is the side opposite angle B, and c is the side opposite angle C: 1) a2 = b2 + c2 – 2bc cos A 2) b2 = a2 + c2 – 2ac cos B 3) c2 = a2 + b2 – 2ab cos C

b. cos(–A) = cosA c. tan(–A) = –tanA 6. Half-angle formulas

TRIG WITH THE UNIT CIRCLE 1. The unit circle is a circle that has a center of (0, 0), the origin, on a rectangular coordinate plane with a radius of one (r = 1); points on the circle have coordinates (x, y). 2. One radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle; in a unit circle, r = 1, so when the arc length, s, is also 1, then the central angle has a measure of one radian. 3. The circumference of the unit circle = 2rπ = 2(1)π = 2π radians = 360°; a semicircle is π radians = 180°. a. 180° = π radians

a. b. c. 7. Double-angle formulas a. sin2A = 2sinA cosA b. cos2A = cos2 A – sin 2 A or cos2A = 1 – 2sin 2 A or cos2A = 2cos2 A – 1 c.

°

⎛ 180 ⎞⎟ ⎟ b. 1 radian = ⎜⎜ ⎜⎝ π ⎟⎟⎠

8. Product-sum formulas

c. d. To

convert

)

degrees

and

radians,

degree measure of the angle . 180°

a.

radian measure of the angle = π radians

b.

4. Trig function definitions using a unit circle with central angle A and the terminal side of angle A intersecting the unit circle at point (x, y): a. Sine of angle A: sin A = y b. Cosine of angle A: cos A = x c. Tangent of angle A:

d.

when x ≠ 0

d. Cosecant of angle A: e. Secant of angle A:

c.

e.

when y ≠ 0

f.

when x ≠ 0

f. Cotangent of angle A:

ISBN-13: 978-142322308-5 ISBN-10: 142322308-X

g.

when y ≠ 0

h.

Author: S. B. Kizlik Disclaimer: This guide is intended for informational purposes only. Due to its condensed format, it cannot $6.95

possibly cover every aspect of the subject. BarCharts, Inc., its writers, editors, and design staff are not responsible or liable for the use or misuse of the information contained in this guide. All rights reserved. No part of this publication may be reproduced or transmitted in any form, or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without written permission from the publisher. Made in the USA © 2002–2014 BarCharts, Inc. 0514 Find us on

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