Beginning and Intermediate Algebra - Wallace Math Courses ...

More documents

Recommendations

Info

$Solving Linear Equations - Distance, Rate and Time - Wallace Math ...$

$Inequalities - Compound Inequalities - Wallace Math Courses$

$Solving Linear Equations - Absolute Value - Wallace Math Courses$

6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials. There are many benifits of a polynomial being factored. We use factored polynomials to help us solve equations, learn behaviors of graphs, work with fractions and more. Because so many concepts in algebra depend on us being able to factor polynomials it is very important to have very strong factoring skills. In this lesson we will focus on factoring using the greatest common factor or GCF of a polynomial. When we multiplied polynomials, we multiplied monomials by polynomials by distributing, solving problems such as 4x 2 (2x 2 − 3x + 8) = 8x 4 − 12x 3 + 32x. In this lesson we will work the same problem backwards. We will start with 8x 2 − 12x 3 + 32x and try and work backwards to the 4x 2 (2x − 3x +8). To do this we have to be able to first identify what is the GCF of a polynomial. We will first introduce this by looking at finding the GCF of several numbers. To find a GCF of sevearal numbers we are looking for the largest number that can be divided by each of the numbers. This can often be done with quick mental math and it is shown in the following example Example 262. Find the GCF of 15, 24, and 27 15 24 27 =5, = 6, =9 Each of the numbers can be divided by 3 3 3 3 GCF =3 Our Solution When there are variables in our problem we can first find the GCF of the num- 212
ers using mental math, then we take any variables that are in common with each term, using the lowest exponent. This is shown in the next example Example 263. GCF of 24x 4 y 2 z, 18x 2 y 4 , and 12x 3 yz 5 24 6 =4, 18 6 =3, 12 6 = 2 Each number can be divided by 6 x 2 y x and y are in all 3, using lowest exponets GCF = 6x 2 y Our Solution To factor out a GCF from a polynomial we first need to identify the GCF of all the terms, this is the part that goes in front of the parenthesis, then we divide each term by the GCF, the answer is what is left inside the parenthesis. This is shown in the following examples Example 264. 4x 2 4 = x2 , − 20x 4 4x2 − 20x + 16 GCF is 4, divide each term by 4 16 = − 5x, =4 4 4(x This is what is left inside the parenthesis 2 − 5x +4) Our Solution With factoring we can always check our solutions by multiplying (distributing in this case) out the answer and the solution should be the original equation. Example 265. 25x 4 5x 2 = 5x2 , Example 266. 25x 4 − 15x 3 + 20x 2 GCF is 5x 2 , divide each term by this − 15x3 20x2 = − 3x, = 4 This is what is left inside the parenthesis 5x2 5x2 5x 2 (5x 2 − 3x + 4) Our Solution 3x 3 y 2 z + 5x 4 y 3 z 5 − 4xy 4 GCF isxy 2 , divide each term by this 213
Page 1 and 2:
Beginning and Intermediate Algebra
Page 3 and 4:
Special thanks to: My beautiful wif
Page 5 and 6:
Chapter 6: Factoring 6.1 Greatest C
Page 7 and 8:
0.1 Pre-Algebra - Integers Objectiv
Page 9 and 10:
Multiplication and division of inte
Page 11 and 12:
45) (4)( − 6) Find each quotient.
Page 13 and 14:
9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
Page 15 and 16:
Example 21. 13 6 − 9 6 4 6 2 3 Sa
Page 17 and 18:
Find each quotient. 37) − 2 ÷ 7
Page 19 and 20:
the expression that is to be evalua
Page 21 and 22:
Solve. 1) − 6 · 4( − 1) 3) 3+(
Page 23 and 24:
It will be more common in our study
Page 25 and 26:
0.4 Practice - Properties of Algebr
Page 27 and 28:
Chapter 1 : Solving Linear Equation
Page 29 and 30:
Addition Problems To solve equation
Page 31 and 32:
8x = − 24 8 8 x = − 3 Division
Page 33 and 34:
1.2 Linear Equations - Two-Step Equ
Page 35 and 36:
− 1 − 1 Divide both sides by
Page 37 and 38:
1.3 Solving Linear Equations - Gene
Page 39 and 40:
Example 63. − 3x +9=6x − 27 Not
Page 41 and 42:
4[ − 2]+9=−15 +8(2) Finish mult
Page 43 and 44:
1.4 Solving Linear Equations - Frac
Page 45 and 46:
Example 72. � � 3 5 4 x + = 3 2
Page 47 and 48:
1.5 Solving Linear Equations - Form
Page 49 and 50:
Example 79. A = πr 2 +πrs for s S
Page 51 and 52:
1.5 Practice - Formulas Solve each
Page 53 and 54:
|x| = 5 Absolute value can be posit
Page 55 and 56:
Example 90. 7+|2x − 5| =4 Notice
Page 57 and 58:
1.7 Solving Linear Equations - Vari
Page 59 and 60:
Once we have found the constant of
Page 61 and 62:
1.7 Practice - Variation Write the
Page 63 and 64:
sure. If the pressure of a certain
Page 65 and 66:
Fifteen more than three times a num
Page 67 and 68:
When we started with our first, sec
Page 69 and 70:
1.8 Practice - Number and Geometry
Page 71 and 72:
30. The perimeter of a college bask
Page 73 and 74:
Our equation comes from the future
Page 75 and 76:
4x + 2=102 Solve the two − step e
Page 77 and 78:
years ago, the bronze plaque was on
Page 79 and 80:
1.10 Solving Linear Equations - Dis
Page 81 and 82:
Rate Time Distance Bob r + 2 3 Fred
Page 83 and 84:
2 =t Our solution fort, she catches
Page 85 and 86:
Find the distance he rode. 8. A man
Page 87 and 88:
The first plan is flying 25 mph slo
Page 89 and 90:
2.1 Graphing - Points and Lines Obj
Page 91 and 92:
E B D F G C A Our Solution The main
Page 93 and 94:
x y − 3 − 4 0 − 2 3 0 Our com
Page 95 and 96:
2.2 Graphing - Slope Objective: Fin
Page 97 and 98:
When mathematicians began working w
Page 99 and 100:
Example 131. Find the value of x be
Page 101 and 102:
9) 10) Find the slope of the line t
Page 103 and 104:
y = − 2 x + 3 Our Solution 3 Iden
Page 105 and 106:
2.3 Practice - Slope-Intercept Writ
Page 107 and 108:
2.4 Graphing - Point-Slope Form Obj
Page 109 and 110:
Find the equation of the line throu
Page 111 and 112:
Write the point-slope form of the e
Page 113 and 114:
m = 2 5 m = 2 5 Example 147. Slope
Page 115 and 116:
2.5 Practice - Parallel and Perpend
Page 117 and 118:
Chapter 3 : Inequalities 3.1 Solve
Page 119 and 120:
than 4. If we have an expression su
Page 121 and 122:
− 5 − 5 − 2x � 6 Divide bot
Page 123 and 124:
Solve each inequality, graph each s
Page 125 and 126:
As the graphs overlap, we take the
Page 127 and 128:
3.2 Practice - Compound Inequalitie
Page 129 and 130:
edition of a college algebra text!
Page 131 and 132:
It is important to remember as we a
Page 133 and 134:
Chapter 4 : Systems of Equations 4.
Page 135 and 136:
ested in the point that is a soluti
Page 137 and 138:
that parallel lines have the same s
Page 139 and 140:
4.2 Systems of Equations - Substitu
Page 141 and 142:
tion. This process is described and
Page 143 and 144:
World View Note: French mathematici
Page 145 and 146:
31) 2x + y = 2 3x + 7y = 14 33) x +
Page 147 and 148:
6x +9y = 6 New second equation −
Page 149 and 150:
Just as with graphing and substutio
Page 151 and 152:
4.4 Systems of Equations - Three Va
Page 153 and 154:
just the ordered pair (x, y). In th
Page 155 and 156:
15x − 12y +9z = − 12 5x − 4y
Page 157 and 158:
23) 3x + 3y − 2z = 13 6x + 2y −
Page 159 and 160:
are given to use. This means someti
Page 161 and 162: − 0.5 − 0.5 c=17 We have c, num
Page 163 and 164: − 0.06x − 0.06y = − 240 Add e
Page 165 and 166: 14) A bank contains 27 coins in dim
Page 167 and 168: 4.6 Systems of Equations - Mixture
Page 169 and 170: Amount Part Total Start 40 3 120 Ad
Page 171 and 172: 4(30 −n)+2.5n=105 Substitute into
Page 173 and 174: 16) A certain grade of milk contain
Page 175 and 176: 175
Page 177 and 178: 5.1 Polynomials - Exponent Properti
Page 179 and 180: A quicker method to arrive at the s
Page 181 and 182: Example 210. Example 211. 7a 3 (2a
Page 183 and 184: 5.2 Polynomials - Negative Exponent
Page 185 and 186: As we simplified our fraction we to
Page 187 and 188: 5.2 Practice - Negative Exponents S
Page 189 and 190: Example 223. Convert 3.21 × 10 5 t
Page 191 and 192: 5.3 Practice - Scientific Notation
Page 193 and 194: World View Note: Ada Lovelace in 18
Page 195 and 196: 21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
Page 197 and 198: Just as we distribute a monomial th
Page 199 and 200: Example 243. (2x − 5)(4x 2 − 7x
Page 201 and 202: 5.6 Polynomials - Multiply Special
Page 203 and 204: Be very careful when we are squarin
Page 205 and 206: 5.7 Polynomials - Divide Polynomial
Page 207 and 208: 3. Change the sign of the terms and
Page 209 and 210: Example 261. 2x 3 + 42 − 4x x + 3
Page 211: Chapter 6 : Factoring 6.1 Greatest
Page 215 and 216: 6.1 Practice - Greatest Common Fact
Page 217 and 218: Example 273. 10ab + 15b + 4a+6 Spli
Page 219 and 220: example the binomials are (a + b) a
Page 221 and 222: 6.3 Factoring - Trinomials where a
Page 223 and 224: As the past few examples illustrate
Page 225 and 226: 6.3 Practice - Trinomials where a =
Page 227 and 228: Example 296. 10x 2 − 27x +5 Multi
Page 229 and 230: 6.5 Factoring - Factoring Special P
Page 231 and 232: Example 306. 4x 2 + 20xy + 25y 2 Mu
Page 233 and 234: 6.5 Practice - Factoring Special Pr
Page 235 and 236: Example 313. Example 314. Example 3
Page 237 and 238: 6.7 Factoring - Solve by Factoring
Page 239 and 240: 0=(2x +3)(2x − 3) One factor must
Page 241 and 242: 6.7 Practice - Solve by Factoring S
Page 243 and 244: 7.1 Rational Expressions - Reduce R
Page 245 and 246: However, if there is more than just
Page 247 and 248: 41) 8m+16 20m − 12 43) 2x2 − 10
Page 249 and 250: Example 333. 25x2 24y4 · 9y8 55x7
Page 251 and 252: Simplify each expression. 1) 8x2 9
Page 253 and 254: 7.3 Rational Expressions - Least Co
Page 255 and 256: As the above example illustrates, w
Page 257 and 258: 7.4 Rational Expressions - Add & Su
Page 259 and 260: 13 12 Our Solution The same process
Page 261 and 262: 7.4 Practice - Add and Subtract Add
Page 263 and 264:
2(12) 1(12) − 3 4 5(12) 1(12) + 6
Page 265 and 266:
LCD =(x + 3)(x − 3) Multiply each
Page 267 and 268:
23) 25) 27) 29) x − 4+ 9 2x +3 x
Page 269 and 270:
(20)(9)=6x Multiply 180 = 6x Divide
Page 271 and 272:
3.5 6 The man hasashadow of 3.5 fee
Page 273 and 274:
33) Kali reduced the size of a pain
Page 275 and 276:
LCD will be more involved. We will
Page 277 and 278:
x=5 or − 2 Our Solution World Vie
Page 279 and 280:
7.8 Rational Expressions - Dimensio
Page 281 and 282:
� 435 1 �� 1 lbs = 16 435
Page 283 and 284:
To focus on the process of conversi
Page 285 and 286:
7.8 Practice - Dimensional Analysis
Page 287 and 288:
Chapter 8 : Radicals 8.1 Square Roo
Page 289 and 290:
√ process is being able to transl
Page 291 and 292:
Simplify. √ 1) 245 √ 3) 36 √
Page 293 and 294:
Example 384. 3√ 54 We are working
Page 295 and 296:
8.3 Radicals - Adding Radicals Obje
Page 297 and 298:
Simiplify 1) 2 5 √ + 2 5 √ + 2
Page 299 and 300:
√ √ √ √ 4 50 + 6 30 − 8 3
Page 301 and 302:
The previous example could have bee
Page 303 and 304:
8.5 Radicals - Rationalize Denomina
Page 305 and 306:
2x2 √ √ 10 − 6x 3x Our Soluti
Page 307 and 308:
Example 408. 2 5 √ − 3 7 √ 5
Page 309 and 310:
37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
Page 311 and 312:
1 (3) 4 Evaluate exponent 1 81 Our
Page 313 and 314:
8.6 Practice - Rational Exponents W
Page 315 and 316:
Example 420. ab2 3√ a2 4√ b Rew
Page 317 and 318:
8.7 Practice - Radicals of Mixed In
Page 319 and 320:
Example 426. Example 427. i 35 Divi
Page 321 and 322:
√ Dividing with complex numbers a
Page 323 and 324:
Simplify. 1) 3 − ( − 8+4i) 3) (
Page 325 and 326:
Chapter 9 : Quadratics 9.1 Solving
Page 327 and 328:
Example 444. + 1 +1 Add1to both sid
Page 329 and 330:
2 − 2=0 Subtract 0=0 True! It wor
Page 331 and 332:
Solve. √ 1) 2x + 3 √ 3) 6x −
Page 333 and 334:
x4 4√ = ± 4√ 16 Simplify roots
Page 335 and 336:
5√ 4x +1 = ± 3 Clear radical by
Page 337 and 338:
9.3 Quadratics - Complete the Squar
Page 339 and 340:
x2 + 5 25 x+ 3 36 � x + 5 �2 6
Page 341 and 342:
Example 464. − 7 3 � 3 3 + 3 3
Page 343 and 344:
9.4 Quadratics - Quadratic Formula
Page 345 and 346:
x = Example 468. √ 30 ± 900 + 11
Page 347 and 348:
9.4 Practice - Quadratic Formula So
Page 349 and 350:
12x 2 − 17x + 6=0 Our Solution If
Page 351 and 352:
9.5 Practice - Build Quadratics fro
Page 353 and 354:
Example 479. a −2 − a −1 −
Page 355 and 356:
12 = (x − 3) 2 or 13 =(x − 3) 2
Page 357 and 358:
9.7 Quadratics - Rectangles Objecti
Page 359 and 360:
Example 486. 2 Our Solution The len
Page 361 and 362:
4(x 2 + 10x − 24) =0 Factor trino
Page 363 and 364:
that require 24 square meters for p
Page 365 and 366:
1(12x) 3 + 1(12x) x 1 1 5 + = 3 x 1
Page 367 and 368:
6x +6(x − 1)=5x(x − 1) Reduce e
Page 369 and 370:
16) A sink can be filled from the f
Page 371 and 372:
These simultaneous product equation
Page 373 and 374:
9.10 Quadratics - Revenue and Dista
Page 375 and 376:
96n(n − 2) 96n(n − 2) +4n(n −
Page 377 and 378:
upstream, the current will pull aga
Page 379 and 380:
12) A pilot flying at a constant ra
Page 381 and 382:
The above method to graph a parabol
Page 383 and 384:
It is important to remember the gra
Page 385 and 386:
Chapter 10 : Functions 10.1 Functio
Page 387 and 388:
Example 504. Which of the following
Page 389 and 390:
2x − 3 � 0 Set up an inequality
Page 391 and 392:
Solve. 10.1 Practice - Function Not
Page 393 and 394:
10.2 Functions - Operations on Func
Page 395 and 396:
(x − 5)(x +1) (x − 5) Divide ou
Page 397 and 398:
We can also evaluate a composition
Page 399 and 400:
23) f(x) = x 2 − 5x g(x) = x + 5
Page 401 and 402:
10.3 Functions - Inverse Functions
Page 403 and 404:
(3x +6)(3 − 4x) 3 − 4x (4x+8)(3
Page 405 and 406:
10.3 Practice - Inverse Functions S
Page 407 and 408:
Example 530. 8 3x = 32 Rewrite 8 as
Page 409 and 410:
10.4 Practice - Exponential Functio
Page 411 and 412:
log3x =7 Identify base, 3, answer,
Page 413 and 414:
10.5 Practice - Logarithmic Functio
Page 415 and 416:
A = 25000(1.01625) 20 Evaluate expo
Page 417 and 418:
The variable e is a constant simila
Page 419 and 420:
j. All of the above compounded cont
Page 421 and 422:
Using the diagram at right, find ea
Page 423 and 424:
10.7 Practice - Trigonometric Funct
Page 425 and 426:
21) 23) 25) 27) A B A 5 38 ◦ x x
Page 427 and 428:
37) 39) A A 37.1 ◦ C x x C 13.1 4
Page 429 and 430:
θ 17 From angle θ the given sides
Page 431 and 432:
function to find the missing angle
Page 433 and 434:
Find the measure of each angle indi
Page 435 and 436:
27) 29) 14 A A θ C 15 7 C 14 θ B
Page 437 and 438:
437
Page 439 and 440:
7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
Page 441 and 442:
54) 60v − 7 55) − 3x + 8x 2 56)
Page 443 and 444:
45) 12 46) All real numbers 47) No
Page 445 and 446:
15) wx 3 = 1458 16) h = 1.5 j 17) a
Page 447 and 448:
1) B(4, − 3) C(1, 2) D( − 1, 4)
Page 449 and 450:
1) y = 2x +5 2) y = − 6x +4 3) y
Page 451 and 452:
23) x =4 24) y − 4 = 7 (x − 1)
Page 453 and 454:
25) 5�x
Page 455 and 456:
1) ( − 2, 4) 2) (2,4) 3) No solut
Page 457 and 458:
27) 56, 144 28) 1.5, 3.5 29) 30 30)
Page 459 and 460:
42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
Page 461 and 462:
5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
Page 463 and 464:
7) (b +8)(b + 4) 8) (b − 10)(b
Page 465 and 466:
11) n(n − 1) 12) (5x +3)(x − 5)
Page 467 and 468:
5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
Page 469 and 470:
7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
Page 471 and 472:
21) 0, 5 22) − 2, 5 3 23) 4,7 24)
Page 473 and 474:
42) − 18xz 4x3yz3 � 4 8.3 1) 6
Page 475 and 476:
16 3 √ + 4 5 √ 43 18) 19) 2 √
Page 477 and 478:
8.8 1) 11 − 4i 2) − 4i 3) − 3
Page 479 and 480:
√ √ 33) 4+i 39, 4 −i 39 34)
Page 481 and 482:
9) ± 2, ± 4 10) 2,3, − 1 ±i 3
Page 483 and 484:
1) 2) 3) 4) 5) 6) (-2,0) (0,-8) (-1
Page 485 and 486:
11) 100 12) − 74 13) 1 5 14) 27 1
Page 487 and 488:
10.5 1) 9 2 = 81 2) b −16 = a 3)
Page 489:
21) 51.3 ◦ 22) 45 ◦ 23) 56.4
show all

Beginning and Intermediate Algebra - Wallace Math Courses ...

Create successful ePaper yourself

Delete template?

Save as template?