# HW 1

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School
University of California, Berkeley **We aren't endorsed by this school
Course
MATH 16A
Subject
Mathematics
Date
Oct 24, 2023
Pages
3
Uploaded by ConstableSteelCheetah21 on coursehero.com
R2 Factoring R-5 CAUTION | Avoid the common error of writing (x + y)? = x> + y2. As the first step of Example 5 shows, the square of a binomial has three terms, so e+ y)2 =x* + 2xy +'y2.'-«\ 3 Furthermore, higher powers of a binomial also result }n more than two terms. For example, verify by multiplication that (x + y)3 =x>+ 3%y + 3xy® + ¥ Remember, for any value of n # 1, (x + y)" #x" + y" R.| Exercises Ferform the indicated operations. 15. 3p )(9p* +3p + 1) 1022 6x + L1) + (=32 + 7x 2) 16. (3p +2)(5p* +p 4) 242 —3y+8) - (P —6y—2) 17. (2m + 1)(4m® 2m + 1) 3 6(2¢° + 4 3) + 4(-¢* + 7q 3) 18. (k + 2)(12K* 3K* + k + 1) 220372 + dr +2) = 3(—rF +4r = 5) 19. (x+y+2)Bx— 2 —2) 5 0613x% 4215x + 0.892) 047(2x* 3x + 5) 20. (r + 2s = 30)(2r 25 + 1) £ 05(572 +32r—6) (1772 = 2r 1.5) 21, (x + 1)(x + 2)(x + 3) = —om(2m® + 3m 1) 22, (x D(x + 2)(x - 3) £ ar =20 + 50+ 6) 23.(x +2)° 83— 2y)(3t + 5y) 24. (2a 4b)? 18 ok + q)(2k q) 25. (x ) 2 —3x)(2 + 3x) 26. (3x +y)* s = 5)(6m 5) YOUR TURN ANSWERS s 1. —9x* + 8x + 13 2.12)° + 29> 19y 10 3.6x° 4+ 19x 7 4. 27x° + 54x%y + 36xy° + 8y* R.2 Factoring Multiplication of polynomials relies on the distributive property. The reverse process, where a polynomial is written as a product of other polynomials, is called factoring. For example, one way to factor the number 18 is to write it as the product 9 - 2; both 9 and 2 are factors of 18. Usually, only integers are used as factors of integers. The number 18 can also be written with three integer factors as 2+ 3. The Greatest Common Factor To factor the algebraic expression 15m + 45, first note that both 15m and 45 are divisible by 15; 15m = 15+ m and 45 = 15+ 3. By the distributive property, 15m + 45 =15-m + 15-3 = 15(m + 3).
R-8 CHAPTERR Algebra Reference R.2 Exercses Factor each polynomial. If a polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary. < AW P = S R = N R S .y = dyz 217 L 3%+ dx 7 . 3a® + 10a + 7 15y +y— 2 . 7d® + 14a? . 3y° + 24y + 9y . 13p*q* 39p°q + 26p°¢* . 60m* 120m*n + 50m*n® m? 5m 14 L+ dx—5 L2+ 92+ 20 L b2 —8b + 7 . @* 6ab + 5b* . 5% + 25t 352 2 15: 16. 17. 18. 19. 20. 21. 23. 25. 27. 29. 31 21m? + 13mn + 202 6a® 48a 120 3m® + 12m* + 9m 4a* + 10a + 6 24a* + 10a°b 4a°h* 24x* + 36x%y 60x%? x2 64 10x* 160 22 + ldzy + 49y* 9p* 24p + 16 27r% 64s° x =yt . 9m? 25 . Ox? + 64 . 52— 10st + 252 . - 216 . 3m® + 375 . 16a* 81p* YOUR TURN ANSWERS mmm 1. 3 22222 + 27 + 9) (2a b)(3a + 4b) R.3 Rational Expressions 2. (x +2)(x - 5) Many algebraic fractions are rational expressions, which are quotients of polynomials with nonzero denominators. Examples include 8 R 3x% + 4x d R Next, we summarize properties for working with rational expressions. Properties of Rational Expressions For all mathematical expressions P, Q, R, and S, with Q # 0 and § # 0: L E Al v:i:: I 2 o © | | ST ST STER Y BERS = 1 = x| e When writing a rational expression in lowest terms, we may need to use the fact that n = g"™". For example, p: Fundamental propert Addition Subtraction Multiplication Division y
= R-16 CHAPTERR Algebra Reference R.4 exeraises Solve each equation. CAUTION | It is possible to get, as a solution of a rational equation, a number that makes one or more of the denominators in the original equation equal to zero. That number is not a solution, so it is necessary to check all potential solutions of rational equations. These introduced solutions are called extraneous solutions. P20V ENSFA Solving a Rational Equation i 20> By x—3 x x{(x—3) SOLUTION The common denominator is x(x 3). Multiply both sides by x(x 3) and solve the resulting equation. Solve x(x—3)-<}%3+l)—x(x—3)'[]fi] 2x+x—3= 3x = x=3 Checking this potential solution by substitution into the original equation shows that 3 makes two denominators 0. Thus, 3 cannot be a solution, so there is no solution for this equation. A T Solve each equation. =x— -2 x+2 L2x+8=x—4 g X2 2.5 _7-6_% 2.5 +2=8—3x 7 5 4 3.02m 05=0.1m + 0.7 29, A 3 =0 '2' '3 1 5 "x—3 "2+ =3 4. —k—k+-== 5 7 12 3 82 = -2 pH2 p'—4 53 +2—5(r+1)=6r+4 om 6 12 6. 564 8) vk A 5= (8 =) B o =k 7.28m 23 —m) —4]=6m 4 2y 5 10— 8y 7 R 8.42p—(3-p)+5]=~Tp-2 PR TN =y 1 3x 2 Solve each equation by factoring or by using the quadratic 33. D R 3%+ 2 formula. If the solutions involve square roots, give both the 5 exact solutions and the approximate solutions to three decimal 34 5 S/ a —2a+4 places. B R a@+a 9.2 +5c+6=0 10. x> = 3 + 2 e P 6 - 2 10 m? = 14m 49 12, 2% k=10 BRI SR i 13. 122 = 5x =2 14. m(m 7) = —10 36. 2 + 3 = 1 i -2 —3 2—x—-6 X+3x+2 15.4x* 36 =0 16. z(2z +7) = 4 P 5 3 17. 12y 48y = 0 19. 2m® 4m =3 21. k2 10k = —20 23.2 = Tr+5=0 25. 3k + k=6 8= B =0 L +p—1=0 37. " o2 o A 2 +3x~9 Wf=%=3 xFLEL3 YOUR TURN ANSWERS = L 52— 8x+2=0 L 22— Tx+30=0 1. -3/2 2.3/2,-5 . 5m* + 5m =0 3.4 V10 4. —1,-4
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