Chapter 8. Lesson a. (2x+3)(x+2) b. (2x+1)(3x+2) c. no solution d. (2x+y)(y+3) ; Conclusion. Not every expression can be factored.

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1 Chapter 8 Lesson a. (x+4)(y+x+) = xy+x +6x+4y+8 b. 18x +9x 8-. a. (x+3)(x+) b. (x+1)(3x+) c. no solution d. (x+y)(y+3) ; Conclusion. Not every expression can be factored a. (3x+1)(x+5)=6x +17x+5 b. (5x )(y+3)=5xy+15x y 6 c. (4x 3)(3x+4)=1x +7x The product of each diagonal is equal. 6x 5 =30x and x 15x =30x Diagonals: part (a) are both 30x, part (b) are both 30xy, part (c) are both 144x. Typical response: The product of one diagonal always equals the product of the other diagonal (x 3)(x+y 4)=x +4xy 11x 6y a. 1x +17x 5 b. 4x 8x x x 3x x 7x 7x x 6x 8-9. a. m =, (0, 1 ) b. m = 3, (0, 7) c. m = 3, (0, 8) d. m =0, (0, ) a. (0, 8); It is the constant in the equation. b. (, 0) and (4, 0); The product of the x-intercepts equals the constant term. c. (1, 9); Its x-coordinate is midway between the x-intercepts a. 1 b. 7.4 c. 4.4 Answer Key 53

2 Lesson a. (5x )(x 7)=10x 39x+14 b. 35x 4x =10x 14 =140x a. (x+3)(x+1) b. One corner should contain 4x, while the other should contain 6x ; (3x+4)(x+). c. Their sum is 7x, and their product is 1x. d. The product 1x should be placed at the top of the diamond problem, 7x at the bottom, and terms 3x and 4x should be in the middle. e. (x+3)(x+) a. One corner contains 6x, and the opposite corner contains 1. b. The product of the x and units terms (in this case, 7x ) goes on top, while the x-term (17x ) goes on bottom. d. (x+3)(3x+4) a. (x+3)(x+6) b. (4x 3)(x+5) c. (x 3)(x 1) d. not factorable because there are no integers that multiply to get 9x (the diagonal of the generic rectangle) and add to get 5x a. (x 6)(x+) b. (x+1) c. (x 5)(x+1) d. (x+4)(3x ) a. x-intercepts ( 1, 0) and (3, 0), y-intercept. (0, 3) b. x-intercept (, 0), no y-intercept c. x-intercepts ( 3, 0), ( 1, 0), and (1, 0), y-intercept (0, ) d. x-intercept (8, 0), y-intercept (0, 0) a. (0, 9); It is the constant in the equation. b. (3, 0) and ( 3, 0) a. (6,9) b. (0,) 8-0. a. x = 10 3 b. all numbers c. c= y= 1 4 x Algebra Connections

3 Lesson a. (x+3) b. (x+3)(x+1) c. not factorable d. (3m+7)(m ) 8-3. a. (3x )(3x+) b. 4x(3x 4) c. (4k 3)(k 1) d. 0( 5m) 8-4. (x 6)(x+1) or (x 3)(4x+) 8-5. See table below Multiply x x+1 x+7 x +5x 14 x +15x+7 3x+1 3x 5x 6x +5x s+s+s+3=51; 1, 4, and 15 cm 8-8. a. 9 units b. 15 units c. 10 units d. 11 un 8-9. a. (k )(k 10) b. (x+7)(3x ) (, 5) c. (x 4) d. (3m+1)(3m 1) y= x a. 5 b. 6 c. 5 or 6 d. 1 4 e. 8 f. 1 4 or 8 Lesson a. (3x ) b. (9m+1)(9m 1) c. (x 4)(x 7) d. (3n+3)(n+) or (n+1)(3n+6) a. Yes, because there are two different arrangements of tiles that build a rectangle. b. Because there is a common factor of 3 in each of the terms of the original expression and in one of the two binomials in either of the two partially factored forms. c. (i) and (iii) both have common factors, so they could have more than one factored form a. 5 b. 5(x +5x 3) c. Yes; 5(x 1)(x+3) a. 5(x+4)(x 1) b. 3x(x+3)(x 5) c. (x+5)(x 5) d. y(x 5)(x+) a. (x+5)(x 1) b. (x 3)(x+) c. (3x+1)(x+4) d. It is not factorable because no integers have a product of 14 and a sum of y= 3 4 x a. in 7 weeks b. Joman will score more with 1170 points, while Jhalil will have a. Michelle is correct. b. ( 4, 0) , 46, 47; x+(x+1)+(x+)= a. b. 3 c. 1 Answer Key 55

4 Lesson a. b. 3 c y= 3x y=3x 5 ; m =3 and b = There is only one line of symmetry. horizontal through the middle a. x-intercepts (, 0) and (0, 0), y-intercept (0, 0) b. x-intercepts ( 3, 0) and (5, 0), y-intercept (0, 3) c. x-intercepts ( 1, 0) and (1, 0), y-intercept (0, 1) d. x-intercept (9, 0), y-intercept (0, 6) a. 6x +x 1 b. 5x 0x+4 Lesson a. Longest: Maggie, Highest: Jen b. Jen. (0, 0) and (8, 0), Maggie. (3, 0) and (14, 0), Imp. (, 0) and (1, 0), Al. (10, 0) and (16, 0); the x-intercepts tell where the balloon was launched and where it landed. c. Jen. (4, 3), Maggie. (8.5, 30.5), Imp. (7, 5), and Al. (13, 7); maximum height You should be able to connect rule table, table graph, graph situation, and table situation a. One way to write the rule is y=(x+1)(x+)+. b. Yes vertex. (4, 9), x-intercepts. (1, 0) and (7, 0), y-intercept. (0, 7) a b. it does not change the value of the number. c. It tells us that a =0. d. 0 for all e. the result is always a. x-intercepts (, 0), ( 4, 0), and (3, 0), y-intercept. (0, 18) b. x-intercepts (3, 0) and (8, 0), y-intercept. (0, 3) c. x-intercept (1, 0) and y-intercept (0, 4) Jen Maggie Imp Al Solution to part (a) a. ( 3, 0) b a. no solution b. (7, ) 56 Algebra Connections

5 Lesson a. No; the y-intercept is not enough information. b. No; the parabola could vary in width and direction. c. Yes; solution shown at right a. y = 0 for all x-intercepts and x = 0 for all y-intercepts. b. (0, 1) c. 0=x +5x 1 d. Not yet, because it has an x term a. At least one of the two numbers must be zero. b. At least one of the three numbers must be zero. c. Typical response: If the product of two or more numbers is zero, then you know that one of the numbers must be zero a. 0 = (x 3)(x+4) b. x 3=0 or x+4 =0, so x = 3 or x = 4. c. The roots are at ( 3, 0) and ( 4, 0). d. The solution graph is shown at right This parabola should have roots ( 3, 0) and (, 0) and y-intercept (0, 6) roots: ( 1, 0) and (, 0), y-intercept: (0, 4) a. One is a product and the other is a sum. b. first: x = or x =1 ; second: x = a. x = or x = 8 b. x =3 or x =1 c. x = 10 or x =.5 d. x = a. The line x =0 is the y-axis, so this system is actually finding where the line 5x y=4 crosses the y-axis. b. (0, ) a. 4; Since the vertex lies on the line of symmetry, it must lie halfway between the x-intercepts. b. (4, ) a. (x )(x+1) b. 4(x 3) a. The symbol represents greater than or equal to and the symbol > represents greater than. b. 5 >3 c. x 9 d. is less than 7. Answer Key 57

6 Lesson The parabola should have y-intercept (0, ) and roots ( 1, 0) and (, 0) a. x = or x = 4 b. x = 1 or x = 4 3 c. x = 5 or x = 3 d. x = 0 or x = 6 e. x = 5 or x = 1.5 f. x = or x = a. y=(x+3)(x )= x +x 6 b. y=(x+5)(x 1)= x +4x By symmetry, (1, 0) is also a root. Thus, the quadratic must be of the form y=a(x )(x 1). Since the parabola points down, a must be negative. Testing a point shows that y= (x )(x 1)= x +14x 4 is correct The result must be the original expression because multiplying and factoring are opposite processes; 65x +1x a. x =3 or x = 3 b. x = or x =5 c. x = 3 or x = d. x = 1 or x = a. true b. false c. true d. true e. false f. false a. 1 b. 1.6 c y= 4 3 x a. 4 3 b. Yes; it makes the equation true and lies on the graph of the line. 58 Algebra Connections

7 Lesson (1) b, () e, (3) a, (4) g, (5) d, (6) i Letter A. The client should order the parabola y=(x 1)(x+6). Letter B. The parabola y=(x 5) should be recommended. Letter C. The parabolay= (x+3)(x ) should be recommended a. y= x(x 8)= x +16x b. y= 3(x 10)(x 16) a. y= x +x 8 b. y= x 6x+9 c. y= x 7x d. y= x 4x m = 1, (0, 4) a. 1.4 and 0.3 b. The quadratic is not factorable a. x = 4 or x = 10 b. x = 8 or x = a. 4 b. 10 c. 8 d. 1.5; They are the same a. (1, 1) b. (, 1 ) Lesson a. The quadratic is not factorable. b. There are two roots (x-intercepts). c. The intercepts are 1.5 and a. a =1, b= 3, c= 7 b. 3± and 1.5; yes a. Graphing and factoring with the Zero Product Property a. x = or 1 3 b. x = 7 or.5 c. x = 0.5 or 0.75 d. no solution a. x = 6 or 7 b. x = 3 or 4 c. x = 0 or 5 d. x = 3 or x = 6 or 7; yes no a. The parabola should be tangent to the x-axis. b. Answers vary, but the parabola should not cross the x-axis y= 1 x line. (a) and (c); parabola. (b) and (d) A and D a. false b. true c. true d. true e. true f. false g. true h. false Answer Key 59

8 Lesson a. (3x )(x+5)=0, x = 3 or 5 b. a =6, b =11, c= 10, x = 3 or 5 c. Yes a. x =5.5 or x = 5.5 b. x = or x = 1 c. x = 3 or x =14 d. x = a. 315 and 315 feet; The bases of the arch are 315 feet from the center. b. 630 feet c. 630 feet; y-intercept a. x = 5 b. x = 1 3 c. x = 1 or 5 3 d. x = ± 3 4 or x = 1 or 6; yes a. y=(x+3)(x 1)= x +x 3 b. y=(x )(x+)= x If x = width, x(x+5)=403 ; width = 13 cm (b) and (c) are solutions a. She solved for x when y=0. b. The y-intercept is (0, 5), and a shortcut is to solve for y when x = 0; y= 3 5 x 5. c. x. (8, 0), y. (0, 1) Lesson a. x = 3 or x = 9 b. x = 17.6 or x = 0.36 c. x = 4 3 or x = 1 d. x =4 e. no solution f. x =.5 or x = a. The Zero Product Property only works when a product equals zero. b. x 3x 4 =0 c. x =4 or x = 1; no a. x = 0 seconds and x =.4 seconds, so it is in the air for.4 seconds. b. at x =1. secs c..88 feet d. The sketch should have roots (0, 0) and (.4, 0) and vertex (1.,.88) If n = # nickels and q = # of quarters, 0.05n+0.5q=1.90, n =q+3, and n = 13, so Daria has 13 nickels a. x = ±0.08 b. x = 9 or 4 c. no solution d. x 1.4 or While the expressions may vary, each should be equivalent to y= x +4x a. x = b. x = 15 c. x = d. all numbers Line L has slope 4, while line M has slope 3. Therefore, line L is steeper D 60 Algebra Connections

Selected Answers for Core Connections Algebra

Selected Answers for Core Connections Algebra Lesson 8.1.1 8-6. (2x 3)(x + 2y 4) = 2x 2 + 4xy 11x 6y +12 8-7. a: 12x 2 +17x 5 b: 4x 2 28x + 49 8-8. a: t(n) = 500 +1500(n 1) b: t(n) = 30!5 n 1 8-9. a: b: