Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)


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1 Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5 a) + a ( b) = 5a + a b 5a + a b= 5 a a+ a b= a (5 a+ b) 7. p + = p + = (p + 8. q 5 = q 5 = (q Section 6. Practice Exercises. (a) product prime (c) greatest common factor (d) prime (e) greatest common factor (GCF) (f) grouping.,,,, 6, 8,, c c+ 5= 5 c 5 c+ 5 = 5( c c+ 6d + d + d = (8 d)( d ) + (8 d)( d) + (8 d)( = 8 d(d + d + 5 x + x = x x + x = x ( x + y y = y y y = y ( y)
2 9 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. t t+ 8t = tt t+ 8 tt = t( t + 8 t) 5 9y + y y = y( y + y r r + r = 7r r r + r r = r (7 r + r) ab + a b = ab + ab( a ) = ab(+ a ) t 9t = (8t + 9t+ ) 6x x 5 = ( (6x + x+ 5 p p = 5 p ( p+ ) 6. 5u v 5uv = 5 uv( u v) 5uv = 5 uv( u v. m m = m ( + m) x y 9x y = 9 x y() 9 x y( y ) = 9 x y( y ) 5 ab + 6ab= ab(5 ab) + ab( = ab(5ab +. pt+ pt + 6 pt = pt(6p pt t). mn + 6mn 9mn = mn( m n m + n) (a) x y 8xy z = 6 xy ( x y z) 7 7 5mp q + m q = mq (5p q + m ) 5+ 7y is prime because it is not factorable. w 5u v is prime because it is not factorable. pq + pq 7 pq = 7 pq (6p + p q ) 8mn mn + mn = mn(n 6 n+ m) 5 t + rt t + r t = t ( t + rt t + r ) u v+ 5u v u + 8uv = uuv ( + 5uv u+ 8 v) 8. (a) x x + 8x= x( x + x x x + 8x= x( x x+ 5 9y + y y = y(y y x 6y z = (7x + 6y + z) 6. a + 5b c = (a 5b + c) 7. (a + 6) b(a + 6) = (a + 6)( b) ( x + yx ( + = ( x + (7 y) 8( vw ) + ( w ) = 8 vw ( ) + ( w ) = ( w )(8v+ 5. tr ( + ) + ( r+ ) = tr ( + ) + ( r+ ) = ( r+ )( t x( x+ + 7 x ( x+ = 7 x( x+ ( + x) = 7 xx ( + 5 y( y ) 5 yy ( ) = 5 yy ( )( y 8a ab+ 6ac bc = a( a b) + c( a b) = ( a b)(a+ c) x + x y+ xy + y = x (x+ y) + y (x+ y) = (x+ y)( x + y )
3 Chapter 6 Factoring Polynomials q+ p+ qr+ pr = ( q+ p) + r( q+ p) = ( q+ p)( + r) 56. xy xz+ 7y 7 z = x( y z) + 7( y z) = ( y z)( x+ 7) x + x+ x+ = x(x+ + (x+ = (x+ (x+ ) y + 8y+ 7y+ = y( y+ ) + 7( y+ ) = ( y+ )(y+ 7) t + 6t t = t( t+ + ( ( t+ = (t ( t+ p p p+ = p( p + ( ( p = (p ( p 6y y 9y+ = y(y + ( (y = (y (y 5a + a a = 5 a( a+ 6) + ( )( a+ 6) = ( a+ 6)(5a ) b + b b = b ( b+ + ( ( b+ 5 = ( b+ ( b 8w + w w 5 = w (w + + ( (w + = (w + (w j k + 5k + j + 5= k( j + + ( j + ab 6ac + b c = ( ab c + ( b c = ( b c)(a wx + 7w x = ( j + (k + = 7 w (x + + ( (x + = (x + (7w p x x 9 p + p = x(9p ) + ( p)(9p ) = (9 p )( x p) 69. ay + bx + by + ax = ay + ax + by + bx = ay ( + x) + by ( + x) = ( a+ b)( y+ x) 7. c+ ay+ ac+ 6y = c+ ac+ 6y+ ay = c( + a) + y( + a) = ( c+ y)( + a) vw + w wv = vw vw + w = vw( w + ( w = ( vw + ( w x + 6m+ + x m= 6m+ x m+ + x = m(6 + x ) + (6 + x ) = ( m+ )(6 + x ) 5x + 5x y + x y+ xy = 5 x(x + xy + x y+ y ) = 5 x( x( x + y ) + y( x + y )) = 5 xx ( + y )(x+ y) ab ab+ ab 6b = ba ( a + 6a = ba ( ( a ) + 6( a )) = ba ( )( a + 6) abx b x ab + b = ( bax bx a+ b) = (( b x a b) ( a b)) = ( ba b)( x p q pq rp q + rpq = pq( p q rp+ rq) = pq(( p q) r( p q)) = pq( p q)( r)
4 96 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual st 8st 6t + 8t = 6( tst s t + t ) = 6( tst ( t ( t ) = 6( tt ( s t ) 5 j j k 5 j k + jk = 5 j(j jk jk + k ) = 5 j( j( j k) k ( j k)) = 5 j(j k)( j k ) 79. P = l + w P = (l + w) 8. P = a + b P = (a + b) 8. S = π r + πrh S = π r( r+ h) 8. A = P + Prt A = P( + rt) ( x + x = x + x (6 y y+ = y y (5 w + w+ = w + w+ 9) 5 ( p p+ = p p Answers may vary. For example: 6x + 9x 88. Answers may vary. For example: y y + 7 y 89. Answers may vary. For example: 6 p q + 8p q p q 9. Answers may vary. For example: 8ab ab Section 6. Practice Exercises. (a) positive. different (c) ( x+ ( x = ( x ( x+ = x + x x = x 7x Both are correct. (d) The factorization ( x 6)( x 6) factored further as ( x+ 6)( x+ ). ( x+ 6)( x+ ) is the complete factorization ab 7ab ab = ab(ab 9ab. tt ( 6( t = ( t ( t ). (x ) + 8 x(x ) = (x )( + x) 5. ax + bx 5a b = x( a+ b) 5( a+ b) = ( a+ b)( x m mx pm+ px = mm ( x) pm ( x) = ( m x)( m p) x + x+ 6 = ( x+ 8)( x+ ) y + 8y+ 8 = ( y+ ( y+ 8) z z+ 8 = ( z 9)( z ). w 7w+ = ( w ( w. z z 8 = ( z 6)( z+. w + w = ( w+ 6)( w ) + + can be
5 Chapter 6 Factoring Polynomials 97. p p = ( p 8)( p+. a a+ 9 = ( a 9)( a 5. t + 6t = ( t+ ( t 6. m m+ = ( m ( m 7. x x+ is prime 8. y + 6y+ 8 is prime w w = w 8w+ 65 = ( w ( w 7t+ t + 7 = t + 7t+ 7 = ( t+ 8)( t+ 9) t+ t + 7 = t + t+ 7 = ( t+ 8)( t+ q + q = q + q = ( q ( q+ 9. n + 8n+ 6 = ( n+ ( n+ = ( n+. v + v+ 5 = ( v+ ( v+ = ( v+. a. c. c. b 5. They are both correct because multiplication of polynomials is a commutative operation. 6. They are both correct because multiplication of polynomials is a commutative operation. 7. The expressions are equal and both are correct 8. The expressions are equal and both are correct 9. It should be written in descending order.. To factor a trinomial, write the trinomial in descending order such as x + bx + c. For all factoring problems, always factor out the GCF from all terms.. x+ x = x x = ( x ( x+ ) y 6 + y = y + y 6 = ( y+ )( y 8) 7. x x 7 = ( x x = ( x ( x+ ) 8. z + z 98 = ( z + z 99) = ( z+ ( z 9) p p + p= 8 p( p 5p+ = 8 p( p ( p 5w 5w + 5w = 5 w ( w 7w+ y z y z + 6y z = y z ( y y+ 6) = 5 w ( w )( w = y z ( y 6)( y 6) or y z ( y 6) t u + 6t u + 9t u = t u ( t + 6t+ 9) = t u ( t+ ( t+ or t u ( t+. x + x = ( x x+ = ( x ( x 6). 5. y y 5 = ( y + y+ = ( y+ ( y+ 7) 5a + 5ax+ x = 5( a ax 6 x ) = 5( a x)( a+ x)
6 98 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual 6. m + m+ n = ( m 5m 6 n ) = ( m+ n)( m 6 n) 6. t+ t = t t = ( t 8)( t c 6c= ( c + c+ ) = ( c+ )( c+ d d = ( d + d + = ( d + ( d + x y 9x y + 6xy = xy ( x 9x + 6) = xy ( x ( x y z + 7 yz + 6z 5 = z ( y + 7y+ 6) 5 = z ( y+ ( y+ p 96 p+ 8 = ( p 8 p+ 7) = ( p 7)( p 5w w 5= 5( w 8w 9) = 5( w 9)( w+ 6. x + x+ = ( x+ ( x+ = ( x + 6. z z+ = ( z ( z = ( z 65. t + 8t = ( t+ )( t ) 66. d + d 99 = ( d + ( d 9) 67. The student forgot to factor out the GCF before factoring the trinomial further. The polynomial is not factored completely, because (x has a common factor of. 68. The student forgot to factor out the GCF before factoring the trinomial further. The polynomial is not factored completely, because (5y has a common factor of ( x ( x+ = x + 9x m + m = ( m m+ = ( m ( m x 6x 8= ( x + x+ 7) = ( x+ 9)( x+ 55. c + 6cd + 5 d = ( c+ 5 d)( c+ d) 56. x + 8xy + y = ( x + 6 y)( x + y) 57. a 9ab+ b = ( a b)( a 7 b) 58. m 5mn+ n = ( m n)( m n) 59. a + a + 8 is Prime 6. b 6a + 5 is Prime 6. q+ q 6= q + q 6 = ( q 7)( q+ 9) (a) 7. (a) ( q 7)( q+ = q + q 7 5x+ x + x 5+ x = x + 9x x + 9x = ( x + x ( x )( x ) = + y + y + y+ 6 = y + y+ 6 y + y+ 6= ( y + 6y+ 8) 7. ( y )( y ) = + + x + x + 9 = ( x + ( x + 9) 7. y + y = ( y + 7)( y w + w 5 = ( w + ( w 75.
7 Chapter 6 Factoring Polynomials p p + = ( p 8)( p. w + 5w = (w ( w , 5, 7, 5 78., 7,, For example, c = 6 8. For example, c = 7 Section 6. Practice Exercises. (a) ( x+ ( x = ( x ( x+. = x 8x+ x = x 5x Both are correct. The factorization ( x ( x ) be factored further as ( x ( x ( x ( x+ is the complete factorization. + can +. 5uv u v + 5u v = 5 uv( v u + 5 uv). mn m n + = m( n ( n = ( n ( m ). 5x xy+ y = 5( x ) y( x ) = (5 y)( x ) a 6a a 8 = 6( a 5a = 6( a 7)( a+ ) b + b = ( b + b = ( b+ 6)( b. y y = (y+ ( y ). a + 7a+ 6 = (a+ ( a+ ) 5. 5x x = (5x+ ( x 6. 7y + 9y = (7y ( y+ ) 7. c 5c = (c+ (c ) 8. 6z + z = (z (z w + 7w= w + 7w = (w ( w+ + p + p= p + p = (p+ (5p ) 5q 6+ 6q = 6q 5q 6 = (q+ )(q 7a + a = a + 7a is prime 6b + b = b + 6b is prime 8+ 7x 8x= 7x 8x+ 8 = (7x ( x ) 8+ 5m m= 5m m 8 = (5m+ )(5m 6. 8q + q = (8q ( q+ 7. 6y + 9xy x = (6y 5 x)( y + x) 8. c 9. b. c. n + n+ = (n+ ( n y 7yz + 6 z = ( y z)( y 6 z) m m 8 = ( m 6m = ( m ( m+
8 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. c c+ 7= ( c c+ = ( c 8)( c. 5 y + y + 6 y = y (y + y+ 6) = y (y+ ( y+ 6) w 5w = ( w+ ( w 8) 6t + 7t = (t+ (t p 9p+ = (p ( p ) u u + u = u (u u+ 6 = u (u ( u 9. m m+ 5 = (m (m = (m. a 5a+ = ( a + 5a = ( a+ 7)( a ). x 7x = ( x + 7x+ = ( x+ ( x+ ) r + r+ 9 = (r+ (r+ = (r+ 5c c+ is prime 7s + s+ 9 is prime 5. 8m + mp+ p = (8m mp p ) = ( m+ p)(m p) x 9xy + y = (x 5 y)(x y) 5 p + pq q = ( p q)(5 p+ q) 6. 6w 55wz+ 5z = (6w + 55wz 5 z ) = ( w+ z)(6w 5 z) m + mn 5 n = (m+ 5 n)( m n) a + 5ab 6 b = (a b)( a+ b) x + x + 9 = ( x + ( x + 9) 7. y + y = ( y ( y + 7) w + w 5 = ( w + ( w p p + = ( p 8)( p.. x 7x 5 = (x + ( x. 5y + y + = (5y + ( y + ) r + 5r = 5(6r + r ) = 5(r+ )(r 6x 8x = (8x 9x ) = (6x+ (x ) s 8 st+ t is prime 6u uv+ 5 v is prime t t 5 = (t (5t+.. z + z 8= ( z z+ 9) = ( z 9)( z 5t + 5t = 5( t t+ 6) = 5( t )( t n + n+ = (8n+ (n+ w + w = (7w (w+ x 6x+ 5 = (6x (x 5. q q+ = ( q 6)( q 7) 65. a a = ( a ( a+ )
9 Chapter 6 Factoring Polynomials b + 6b 7 = ( b+ 7)( b x + 9xy + y = ( x + 5 y)( x + y) p pq+ 6 q = ( p 9 q)( p q) a + ab+ b = ( a+ b)( a+ b) x 7xy 8 y = ( x 8 y)( x + y) t t+ = ( t 7)( t z 5z+ 6 = ( z ( z 6 5 5d + d d = d(5d + d y y + y= y(y y b b 8b= b( b b ) = bb ( + ( b w + w+ = ( w + w+ = ( w+ 7)( w+ x y xy + y = y ( x x+ = y ( x ( x pq pq + q = q( p p+ = q ( p ( p u u + u = u(6u + u = u(u )(u+ 8. (a) ft. 8. (a) 85. (a) h= 6t + t+ () () = = = 6 The height of the rock after sec is 6 6t + t+ = ( t t = ( t+ ( t ) ( )( ) ( )( ) ( )( ) h= t+ 5 t = + 5 = 9 = 6 Yes, the result is the same. h= 6t 8t+ ( ) ( ) = = = The height of the ball after sec is ft. 6t 8t+ = 8( t + t = 8 ( t ( t+ ( )( ) ( )( ) ( )( ) h= 8t 5 t+ = = 8 5 = Yes, the result is the same. x x = ( x ( x+ ) x x+ = ( x 6)( x 8. 8z + 5z + z = z (6z 5z 8. = z (z (z+ 8x + x + = (x + (x + 6y 5y = (y (y (a) 87. (a) 88. (a) x x = ( x ( x+ ) x x+ = ( x ( x x 5x 6 = ( x 6)( x+ x 5x+ 6 = ( x )( x x x+ 9 = ( x 9)( x
10 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual x + x+ 9 = ( x+ 9)( x+ 5. w 9w+ = w 8w w+ = ww ( ) ( w ) = ( w )(w Section 6. Practice Exercises. (a) ( 5x+ ( x = ( x ( 5x+ = x 5x+ x = x x Both are correct. The factorization ( x ( x 6) be factored further as ( x ( x ) ( x+ ( x ) is the complete + can +. factorization.. 5x(x ) (x ) = (x )(5x ). 8(y + + 9y(y + = (y + (8 + 9y). 6ab + b a 8 = 6( ab + b a 8) = 6( ba ( + ( a+ ) = 6( a+ ( b ) 5. and 6. and 7. 8 and 8. and 9. 5 and. and. 9 and. and 6.. x + x+ = x + x+ x+ = ( xx+ + ( x+ = ( x+ (x+ y + 7y+ 6= y + y+ y+ 6 = y(y+ + (y+ = (y+ ( y+ ) 6. p p = p p+ p = p( p ) + ( p ) = ( p )(p+ 7. x + 7x 8= x + 9x x 8 = xx ( + 9) ( x+ 9) = ( x+ 9)( x ) y y y y y 6 = + = yy ( + ( y = ( y ( y+ m + 5m = m + 6m m = mm ( + ( m+ = ( m+ (m 6n + 7n = 6n + 9n n = n(n+ (n+ = (n+ (n 8k 6k 9= 8k k + 6k 9 = k(k + (k = (k (k +. 9h h = 9h 6h+ h = ( h h ) + (h ) = (h )(h+.. k k + 5 = k k k + 5 = k(k 5(k = (k (k = (k 6h + h+ 9 = 6h + h+ h+ 9 = h(h+ + (h+ = (h+ (h+ = (h +
11 Chapter 6 Factoring Polynomials 5. Prime 6. Prime 8. 8t t t = t(t t = ( t t ( t+ ) 7. 9z z+ = 9z 5z 6z+ = ( z z (z = (z (z ) a + 5a + 6 = ( a + ( a + ) 9. y y 5 = ( y 7)( y x + x = x + 6x x = xx ( + ( x+ = ( x+ (x y + 8yz 5z = y + 8yz yz 5z = 6 y(y+ z) 5 z(y+ z) = (y+ z)(6y 5 z) a + ab 9 b = (5a ( b a+ b y + 5y+ = (7 y + 5y+ = (7 y y y = (7 yy ( + + ( y+ ) = ( y+ (7 y+ + w+ w = (w + 5w = (w w+ 8w = [ w(w + (w ] = (w ( w+ 5w + w+ 5 = (5w w = (w (5w+ 6z + z+ 5 = (6z z = (8z+ (z x + xy 8y = (x 5xy + y ) = (x y)( x y) 6x x 5 = (x (x +. 8t + t = (t + (t.. 8p + 7p 5 = (8p ( p + a + a + = (a + 7)( a + ) p 9 p+ = p 5p p+ = 5 p(p (p = (p (5 p p + 5pq 6q = p + 8pq pq 6q = p( p+ q) q( p+ q) = ( p+ q)(p q) 6u 9uv+ v = 6u 5uv uv+ v = u(u 5 v) v(u 5 v) = (u 5 v)(u v) 5m + mn n = 5m + 6mn 5mn n = m(5m+ n) n(5m+ n) = (5m+ n)( m n) a + ab 5b = a + 5ab ab 5b = a(a+ 5 b) b(a+ 5 b) = (a+ 5 b)( a b) 6. 6p pq 9q = (p + 7pq+ q ) = ( p+ q)( p+ q) 5. r rs s = r + 6rs 7rs s = ( rr+ ) s 7( sr+ ) s = ( r+ s)(r 7 s) 7. 8y + 6y + y= 6 y(y + y+ 7) = 6 y(y+ 7)( y+
12 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual 5. h + 9hk k = h + hk hk k = hh ( + 7 k) kh ( + 7 k) = ( h+ 7 k)(h k) u + uv 5v = u + 6uv 5uv 5v 5. = ( uu+ v 5( vu+ v = ( u+ v)(u 5 v) 5. Prime t t 6= t t+ t 6 = tt ( + ( t = ( t ( t+ ) m + m 6= m + m m 6 = mm ( + ( m+ = ( m+ ( m ) 6. v + v+ 5 prime 5. Prime z z+ = 6z 8z 6z+ = 8( z z (z = (z (8 z 6w + w+ = 6w + 8w+ w+ = 8 w(w+ + (w+ = (w+ (8w+ b 8b+ 6= b b b+ 6 = bb ( ( b = ( b ( b = ( b q q+ = q q q+ = qq ( ( q = ( q ( q = ( q 5x + 5x = 5( x 5x+ 6) = 5( x x x+ 6) = 5( xx ( ( x ) = 5( x ( x ) a + a 8= ( a a+ 9) = ( a a 9a+ 9) = ( aa ( 9( a ) = ( a ( a 9) x x prime 7x + 8x = (6x + 9x = (6x + x x = ( x(x+ (x+ ) = (x+ ( x y 78y 8 = (y 9y = (y y+ y = ( yy ( + ( y ) = ( y ( y+ p 6p 7 p= p( p 6p 7) 5 = p( p 9p+ p 7) = p( p( p 9) + ( p 9)) = p( p 9)( p+ w w + 8 w = w ( w w+ 8) = w ( w 7w w+ 8) = w ( w( w 7) ( w 7)) = w ( w 7)( w x + x + 7 x= x(x + x+ 7) = x(x + x+ 7x+ 7) = x( x( x+ + 7( x+ ) = x(x+ 7)( x+
13 Chapter 6 Factoring Polynomials 5 7. r + r r = r(r + r = r(r + 8r 5r = r( r( r+ ) 5( r+ )) = rr ( + )(r 77. n n+ 9 = ( n + n = ( n + 6n 5n = ( nn ( + 6) 5( n+ 6)) = ( n+ 6)( n p 8p + p = p( p 9 p+ 6) = p( p p 5p+ 6) = p( p( p 5( p ) = p( p ( p q q 8q= q( q q ) x y + x y+ x = x ( y + y+ = x ( y + y+ y+ = x ( y( y+ + ( y+ ) = x ( y+ ( y+ ab + ab + b = b ( a + a+ = b ( a + a+ a+ = b ( a( a+ + ( a+ ) = b ( a+ ( a+ = qq ( 5q+ q ) = qqq ( ( + ( q ) = qq ( ( q k 7k = ( k + 7k + = ( k + 5k + k + = ( kk ( + + ( k+ ) = ( k + ( k + ) m 5m+ = ( m + 5m = ( m m+ 7m = ( mm ( ) + 7( m )) = ( m )( m+ 7) h + 8h 9= ( h h+ = ( h 5h 9h+ = ( hh ( 9( h ) = ( h ( h 9) x 7x + = ( x ( x ) m + m + = ( m + ( m + 7) No. (x + contains a common factor of. 8. No. (5x contains a common factor of (a) c= x + x 7 = ( ) + ( ) 7 = + 7 = 8 There are 8 customers. x + x 7= ( x x+ 6) = ( x 8)( x ) ( )( ) ( )( ) ( )( ) c= x 8 x = 8 = 8 8 = 8 Yes, the result is the same. 8. (a) i= d + d + = ( ) + ( ) + = = 5 His income is $5.
14 6 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual 85. (a) 86. (a) d + d + = ( d 5d ( d )( d ) = + 5 i= ( d )( d + = ( )( + = ( 8)( 7) = 5 Yes, the result is the same. s = n + n = ( 6) + 6 = = n + n= n( n+ s= n( n+ = 66 ( + = 67 ( ) = Yes, the result is the same. n + n + n s = 6 ( ) + ( ) + = = 6 8 = 6 = ( + + ) n + n + n n n n = 6 6 n n = 6 n( n+ ( n+ s = 6 ( + ( + = 6 5 ( )( 9) = 6 8 = 6 = ( + ( n+ Yes, the result is the same. Section 6.5 Practice Exercises. (a) difference; ( a+ b)( a b).. sum (c) is not (d) square (e) ( a+ b) ; ( a b) x + x = (x ( x+ ) 6x 7x+ 5 = (x (x ab+ ab= ab( + a) 6 5x y xy = 5 xy (x y ) 5pq+ p pq p = p(5pq+ p q = p(5 p( q+ ( q+ ) = pq ( + (5p 7. ax + ab 6x 6 b = a( x + b) 6( x + b) = ( x+ b)( a 6)
15 Chapter 6 Factoring Polynomials x+ 5+ x = x 6x+ 5 = ( x ( x 8. y x = ( y x)( y + x) 9. 6y + y = y + 6y = ( y+ ( y 9. 5p q = (5 pq (5 pq +. a + 7a + is Prime.. x n 5 = x 5 = n 9.. 8st = (9st (9st+ c = () c = c+ c (p q)(p+ q) = p 9q (7x y)(7x+ y) = 9x 6y x 6 = ( x+ 6)( x 6) r 8 = ( r+ 9)( r 9) w = ( w = ( w+ ( w t 9 = ( t+ 7)( t 7) a b = ( a) ( b) = (a+ b)(a b) 9 x y = ( x) y = ( x+ y)( x y) 9m 6 n = (7 m) ( n) = (7m+ n)(7m n) a 9 b = ( a) (7 b) = (a+ 7 b)(a 7 b) 9q + 6 is prime 6 + s is prime y z = ( y+ z)( y z) b c = ( b+ c)( b c) a b = ( a+ b )( a b ) z = ( z) = z+ z 5 t = (5 6 t ) = 5 ( t) = (5 + t)(5 t) 6 7h = 7(9 h ) = 7( + h)( h) z 6 = ( z + ( z = ( z + ( z+ )( z ) 6. y 65 = ( y + ( y = ( y + ( y+ ( y 7. 8 a = ( 9 a )( 9 + a ) ( a)( a)( 9 a) = z = ( z )( + z ) = ( z)( + z)( + z ) 9. x + 5x 9x 5 = x ( x+ 9( x+ ( x ) = ( x+ 9 = ( x+ ( x+ ( x. y + 6y y = y ( y+ 6) ( y+ 6) = ( y+ 6)( y = ( y+ 6)( y+ )( y )
16 8 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual.. c c 5c+ 5 = c ( c 5( c ( c ) = ( c 5 = ( c ( c+ ( c t + t 6t = t ( t+ ) 6( t+ ) ( t ) = ( t+ ) 6 = ( t+ )( t+ ( t. x 8 + x y 9y = ( x 9) + y( x 9) = ( x 9)( + y) = ( x + ( x ( + y) a 5+ a b b= 5( a + b( a x y 9x y + 6 = x ( y 9) ( y 9) = ( a (5 + b) = ( a+ ( a (5 + b) = ( y 9)( x = ( y+ ( y ( x+ )( x ) w z w 5z + 5 w ( z 5( z ( z ( w = = = ( z+ ( z ( w+ ( w y 8y+ 6= y (( y) + = ( y 5z z+ = (5 z) (5 z)() + = (5z ) 6 p + 6 p+ 5 = (6 p) + (6 p)( + 5 9a + ab+ 9b = (6 p + = (7 a) + (7 a)( b) + ( b) = (7a+ b) 5m mn+ 9n = (5 m) (5 m)( n) + ( n) = (5m n) y+ y + = y y+ = ( y + w w= w w+ = ( w ) 8z + zw+ 5w = 5(6z + zw+ 9 w ) = 5(( z) + ( z)( w) + ( w) ) = 5(z+ w) 7. (x+ = 9x + x (y 7) = y 8y (a) x + x+ is a perfect square trinomial 6. 6 p pq+ q = (9 p 6 pq+ q ) = (( p) ( p)( q) + q ) = ( p q) x + x+ = ( x+ ) ; x + 5x+ = ( x+ ( x+ 5. (a) x + x+ 6 is a perfect square trinomial x + x+ 6 = ( x+ 9)( x+ ; x + x+ 6 = ( x+ 6) 5. x + 8x+ 8 = x + (9)( x) + 9 = ( x+ 9) 6. 9y + 78y+ 5 = (y+ (y+ 6. y + y+ 9 = (y+ 9)(y+ 6. a a+ 5= ( a a+ = ( a 6. t t t t = ( ) = ( t +
17 Chapter 6 Factoring Polynomials x + x+ 9 is prime 66. c c+ 6 is prime x + xy + y = ( x) + ( x) y + y = ( x+ y) y + yz + z = ( y) + ( y) z + z = ( y+ z) x 6x + x= x( x x+ = ( xx = ( xx ( x ) y + y + 5y = 5 y(y + y+ = 5 y(y+ (y+ = 5 y(y+ ( y 9 = (( y + (( y = yy ( 6) (a) (a b = + = (a 5+ b)(a 5 b) [(a b][ (a b] (k + 7) 9m = = (k m)(k m) 8. (a) [(k 7) 7 m][ (k 7) 7m] a (area of outer square) b (area of inner square) Total area = a b a b = ( a+ b)( a b) g (area of outer square) h (area of inner square) total area = g h g h = ( g + h)( g h) ( x ) = (( x ) + )(( x ) ) = xx ( ( p+ 6 = (( p+ + 6)(( p+ 6) = (p+ 7)(p (q+ 5 = ((q+ + ((q+ = (q+ 8)(q ) = ( q+ ) (q = 8( q+ )(q 6 ( t + ) = ( + ( t+ ))( ( t+ )) = ( + t+ )( t ) = ( t+ 6)( t+ ) or ( t+ 6)( t ) 8 ( a + = (9 + ( a+ )(9 ( a+ ) = (9 + a+ (9 a = ( a+ ( a+ or ( a+ ( a Section 6.6 Practice Exercises. (a) sum; cubes difference; cubes.. (c) ( a b)( a + ab+ b ) (d) ( a+ b)( a ab+ b ) 6 6x = 6( x ) = 6( x)( + x) 5t = 5( t ) = 5( t)( + t). ax + bx + 5a + 5 b = x( a + b) + 5( a + b) = ( a+ b)( x+ 5. t+ u+ st+ su = ( t+ u) + s( t+ u) = ( t+ u)( + s) 6. 5y + y 6 = (5y )( y+
18 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual v + 5v = (v ( v+ ab 6ab + ab = 8 ab(5ab b+ a) c c 5 = ( c + c+ = ( c+ z + 6z 9 = ( z 6z+ 9) = ( z x,8, y,7 q, w, r s 9 5 z,8, y,7 a,. t,, 7,., z, 6 ab, 5, w, 6 y 9 p,, 8 y 8= y = ( y )( y + y+ x + 7 = x + = ( x+ ( x x+ 9) p = ( p)( + p+ p ) q + = ( q+ ( q q+ w + 6 = w + = ( w+ ( w w+ 6) b 5 = (6 b) 5 = (6b (6b + b+ n = n = n n + n m = + ( m ) 7 = + m m + m 9 5x + 8 y = (5 x) + ( y) = (5x + y)(5x xy+ y ) 7t + 6 u = ( t) + ( u) = (t+ u)(9t tu+ 6 u ) x = ( x ) = ( x + )( x ) b 5 = ( b ) 5 = ( b + ( b a + 9 is prime. w + 6 is prime. t + 6 = t + = ( t+ ( t t+ 6). 8 t = t = ( t)(+ t+ t ) 6. u + 7 = u + = ( u+ ( u u+ 9) x y = x ( y) = ( x y)( x + xy+ y ) 8r 7 t = ( r) ( t) = (r t)(r + 6rt+ 9 t ) 6t + = ( t) + = (t+ (6t t+ 5r + = (5 r) + = (5r+ (5r 5r+ a + 7 = ( a) + = (a+ (a a+ 9) g is prime. h 5 is prime. b + 8 = ( b + 7) = ( b+ ( b b+ 9) c = ( c 8) = ( c )( c + c+ 5 p 5 = 5( p = 5( p+ ( p q 8 = ( q = ( q + )( q )
19 Chapter 6 Factoring Polynomials h = ( h) 6 = h + h+ h k = + ( k ) 5 5 = + k k + k x p 6 = ( x ) = ( x + ( x = ( x + ( x+ )( x ) 8 = ( p ) 9 = ( p + 9)( p 9) = ( p + 9)( p+ ( p x x w = w 9 x x = w + w 9 5w z = 5 w ( z ) 5. ( ) 6 ( 5w z )( 5w 5wz z ) = + + x + x x = x (x+ (x+ 5. = (x+ ( x = (x+ ( x+ ( x x + x x = x (x+ (x+ = (x+ ( x = (x+ ( x+ )( x ) 6 x y = ( x + y )( x y ) = ( x + y )( x+ y)( x y) t = ( + t )( t ) = ( + t )( + t)( t) 8y 6 = (9y + (9y = (9y + (y+ )(y ) u 56 u = u( u 6 ) = uu ( + 6)( u 6) = uu ( + 6)( u+ ( u y 6 y x = x 5 5 y y = x + x 5 5 q 6 6 = ( q ) 8 = ( q + 8)( q 8) = ( q+ )( q q+ ( q )( q + q+ a 6 = ( a ) = ( a + ( a = ( a+ ( a a+ ( a ( a + a x + 6 y = ( x ) + ( y) 6 = ( x + y)( x x y+ 6y ) a + b = a + ( b ) 6. 6 = ( a+ b )( a ab + b ) u v = ( u ) v = ( u v)( u + u v+ v ) 6. x y = ( x + y )( x y ) = ( x + y )( x+ y)( x y) 6. a b = ( a + b )( a b ) = ( a + b )( a+ b)( a b) 6. k + k 9k 6 = k ( k + 9( k + = ( k + ( k 9) = ( k + ( k + ( k
20 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual w w w+ 8 = w ( w ) ( w ) 6. = ( w )( w = ( w )( w+ )( w ) = ( w+ )( w ) 65. t t t+ = t ( t ( t = ( t ( t = ( t ( t+ ( t 7. (a) + x+ x x + x + x 8 x ( x x ) x + x (x x) x 8 (x 8) a + 7a a = 9 a ( a+ ( a+ = ( a+ (9a = ( a+ (a+ )(a ) 6 p q 5 8 = p q 5 6 = p q p + pq+ q r + s 7 = r + s = r+ s r rs+ s 5 9 a + b = ( a ) + ( b ) a 9 9 b 8 8 = ( a + b )( a a b + b ) = ( a ) ( b ) 6 6 = ( a b )( a + a b + b ) 6 6 = ( a b)( a + ab+ b )( a + a b + b ) 7. (a) x p 75. x y 5 x 8 = ( x )( x + x+ y y+ 9 y+ y + y + y+ 7 ( y + y ) y + y ( y 9 y) 9y + 7 (9y + 7) y + 7 = ( y+ ( y y+ 9) + x+ 5p+ 5 Problem Recognition Exercises. A prime factor cannot be factored further.. Factor out the GCF.. Look for the difference of squares: a b, a difference of cubes: a b, or a sum of cubes: a + b. Grouping 5. (a) Difference of squares
21 Chapter 6 Factoring Polynomials a 6 = ( a 8 = ( a+ 9)( a 9) 6. (a) Nonperfect square trinomial y + y+ = ( y+ ( y+ 7. (a) None of these 6w 6w= 6 w( w 8. (a) Difference of squares 6z 8 = (z + 9)(z+ (z 9. (a) Nonperfect square trinomial t + t+ = (t+ ( t+. (a) Sum of cubes 5r + 5 = 5( r + = 5( r+ ( r r+. (a) Four termsgrouping ac + ad bc bd = a( c+ d) b( c+ d) = ( c+ d)( a b). (a) Difference of cubes x + 5 = ( x+ ( x 5x+. (a) Sum of cubes y + 8 = ( y+ )( y y+. (a) Nonperfect square trinomial 7p 9p+ = (7p ( p 5. (a) Nonperfect square trinomial q 9q = ( q q = ( q ( q+ 6. (a) Perfect square trinomial x + 8x 8= ( x x+ 7. (a) None of these = ( x ) 8a + a= 6 a(a+ ) 8. (a) Difference of cubes 5 y = (7 y ) = ( y)(9 + y+ y ) 9. (a) Difference of squares t = ( t = ( t+ ( t. (a) Nonperfect square trinomial t t 8 = (t+ ( t 8). (a) Nonperfect square trinomial c + c+ = ( c + c+. (a) Four termsgrouping xw x+ yw 5y = xw ( + yw ( = ( w (x+ y). (a) Sum of cubes x +. = ( x+.( x.x+.. (a) Difference of squares q 9 = (q+ (q 5. (a) Perfect square trinomial 6 + 6k + k = 8 + (8)( k) + k = (8 + k) 6. (a) Four termsgrouping
22 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual st+ 5t+ 6s + = ts ( + + 6( s + = ( s + ( t+ 6) 7. (a) Nonperfect square trinomial 6x x 5= (x 7x = (x+ ( x 7. (a) Four termsgrouping x + x xy y = xx ( + yx ( + = ( x+ ( x y) 8. (a) Sum of cubes w + y = ( w+ y)( w wy+ y ) 9. (a) Difference of cubes a c = ( a c)( a + ac+ c ). (a) Nonperfect square trinomial y + y + is prime. (a) Nonperfect square trinomial c + 8c + 9 is Prime. (a) Perfect square trinomial a + a+ = ( a+. (a) Perfect square trinomial b + b+ 5 = ( b+. (a) Nonperfect square trinomial t t+ = ( t + t = ( t+ 8)( t 8. (a) Nonperfect square trinomial y y+ = (y 7 y+ = (5y ( y 9. (a) None of these 5a bc 7 abc = abc (5ac 7). (a) Difference of squares 8a 5= (a = (a (a+. (a) Nonperfect square trinomial t + t 6 = ( t 7)( t+ 9). (a) Nonperfect square trinomial b + b 8 = ( b 8)( b+. (a) Four termsgrouping ab + ay b by = ab ( + y) bb ( + y) = ( b+ y)( a b). (a) None of these 5 6x y + x y = x y ( x+ y) 5. (a) Nonperfect square trinomial 5. (a) Nonperfect square trinomial u uv+ v = (7u v)( u v) p 5p p= p( p + 5p+ = p( p+ ( p+ 6. (a) Difference of squares x y 9 = ( xy + 7)( xy 7) 6. (a) Nonperfect square trinomial 9 p 6 pq+ q is prime 7. (a) Nonperfect square trinomial q 8q 6= (q q
23 Chapter 6 Factoring Polynomials 5 8. (a) Nonperfect square trinomial 58. (a) Nonperfect square trinomial 9w + w 5= (w + w q + q 7 is prime 9. (a) Sum of squares 9m + 6 n is prime 5. (a) Perfect square trinomial 5b b+ 5= 5( b 6b+ 9) = 5( b 5. (a) Nonperfect square trinomial 6r + r+ = (r+ (r+ 5. (a) Nonperfect square trinomial s + s 5 = (s (s+ 5. (a) Difference of squares 6a = (a + (a 5. (a) Four termsgrouping = (a + (a+ (a p + p c 9p 9c = p ( p+ c) 9( p+ c) = ( p+ c)( p 9) = ( p+ c)( p+ ( p 55. (a) Perfect square trinomial 8u 9uv+ 5v = (9 u) (9 u)(5 v) + (5 v) = (9u 5 v) 56. (a) Sum of squares x + 6= ( x (a) Nonperfect square trinomial x 5x 6 = ( x 6)( x+ 59. (a) Four termsgrouping ax 6ay + bx by = ax ( y) + bx ( y) = ( x y)( a+ b) 6. (a) Nonperfect square trinomial 8m m m= m(8m m = m(m+ (m 6. (a) Nonperfect square trinomial x y+ x y+ x y = x y(x + x+ = x y(7x+ )(x+ = ( p+ c)( p+ ( p 6. (a) Difference of squares m 8 = ( m 6 = ( m + 8)( m 8) 6. (a) Four termsgrouping 8uv 6u + v 9 = u(v + (v = (v (u+ 6. (a) Four termsgrouping t t+ st 5s= t( t + s( t = ( t ( t+ s) 65. (a) Perfect square trinomial x x+ = (x x+ = ([ x] ( x)( + ) = (x 66. (a) Perfect square trinomial p + pq+ q = ( p+ q)
24 6 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual 67. (a) Nonperfect square trinomial 6n + 5n n = n(6n + 5n = n(6n n+ 8n = n[ n(n + (n ] = n[(n (n+ ] = n(n (n+ 68. (a) Nonperfect square trinomial k + k k = k(k + k = k(k k + 6k = k[ k(k + (k ] = k[(k (k + ] = k(k (k (a) Difference of squares 6 y = 8 y = (8 y)(8 + y) 7. (a) Difference of squares 6 b b = b(6 b ) = b(6 b ) = b(6 b)(6 + b) 7. (a) Nonperfect square trinomial b b+ is prime. 7. (a) Nonperfect square trinomial y + 6y+ 8 = ( y+ ( y+ ) 7. (a) Nonperfect square trinomial c c + = ( c ( c ) Section 6.7 Practice Exercises. (a) quadratic. 6a 8 ab+ b= (a b(a = (a ( b) b b+ = ( b b+ = ( b 6)( b 8u v uv = uv(uv x + x 8 = (x )( x+ h 75 = ( h = ( h+ ( h x + 6y = ( x + y ) 8. Linear 9. Neither. Quadratic. Quadratic. Neither. Linear. (x (x + = x 5= or x+ = x= 5 x= 5. (x + (x = x+ = or x = x= x= 6. (x )(x + ) = x = or x+ = x= x= x= x= 7. (x 7)(x + 7) = x 7= or x+ 7= x= 7 x= x= x= ;
25 Chapter 6 Factoring Polynomials 7 8. ( x 7)( x 7) = x 7= x = 7 9. ( x+ ( x+ = x + 5= x = 5. (x )(x = x = or x = x= x= x= x=. x(5x = x= or 5x = x= or 5x= x= or x= 5. x(x + 8) = x= or x+ 8= x= or x= 8 x= or 8 x=. The polynomial must be factored completely before applying the zero product rule.. The equation must have one side equal to zero and the other side factored completely p p 5= ( p+ ( p = p+ = or p 5= p= p= 5 y 7y 8= ( y 8)( y+ = y 8= or y+ = y = 8 y = z + z = ( z+ ( z ) = z+ = or z = z = z = w w+ 6 = ( w 8)( w ) = w 8= or w = w= 8 w= q 7q = (q+ ( q = q+ = or q = q= q= x x = (x+ ( x = x+ = or x = x= x= = 9x = (x+ )(x ) x+ = or x = x= x= a 9= (a+ 7)(a 7) = a+ 7= or a 7= 7 7 a= a= k 8k + 96= ( k k + 8) = ( k 6)( k 8) = k 6= or k 8= k = 6 k = 8 = t + t+ 5 ( t + t+ = ( t+ ( t+ = t + 5= t = 5
26 8 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual = m 5m m m(m 5m = m(m+ ( m = m= or m+ = or m = m= m= m= n + n + n= n(n + n+ = n(n+ ( n+ = n= or n+ = or n+ = n= n= n= 7. (p+ ( p ( p+ 6) = p+ = or p = or p+ 6= p= or p= or p= 6. xx ( + 7)(x = x= or x+ 7 = or x 5 = x = or x = 7 or x = 5 5 x = x= or x= 7 or.. x 6x= xx ( 6) = xx ( + ( x = x= or x+ = or x = x= or x= or x= t 6t = tt ( 6) = tt ( + 6)( t 6) = t = or t+ 6= or t 6= t = or t = 6 or t = 6 8. (x (x (x + 7) = x = or x = or x+ 7= x= or x= or x= 7 9. x(x (x + = x= or x = or x+ = x= or x= or x= x = x = or x = or. x(x+ ( x+ = x= or x+ = or x+ = x= or x= or x= x = x= or or x=. 5 x(x+ 9)( x = 5x= or x+ 9= or x = x = 5 or x = 9 or x = 9 x = x= or or x= x + 8x= xx ( + 6) = x= or x+ 6= x= or x= 6 y y= yy ( = y = or y = y = or y= 6m = 9 6m 9 = (m+ (m = m+ = or m = m= or m= 9n = 9n = (n+ (n = n+ = or n = n= or n=
27 Chapter 6 Factoring Polynomials 9 9. y + y = y y + y + y= yy ( + 7y+ = yy ( + ( y+ ) = y = or y+ 5= or y+ = y = or y= 5 or y= 55. b + b + b= bb ( + b+ = bb ( + )( b+ ) = b = b b b= or b+ = or b+ = b= or b= or b= 5. d 6d = d d 6d d = dd ( d 8) = dd ( ( d+ ) = d = or d = or d + = d = or d = or d = 56. x + 6x= x x x + 6x= xx ( x+ 6) = xx ( 6)( x 6) = x= or x 6= or x 6= x= or x= 6 or x= t ( t 7) = 5t t+ = t + = t = 5. 8h= 5( h 9) + 6 8h= 5h 5+ 6 h + 9= h = ( cc 8) = c 6c+ = ( c 8c+ = ( c ( c = c 5= or c = c= 5 or c= qq ( = q 9q = ( q q = ( q ( q+ = q = or q+ = q= or q= ( a + a) = a 9 a + 6a= a 9 a + 6a+ 9= ( a+ ( a+ = a + = a = 9( k = k 9k 9= k k + 9k 9= (k ( k + = k = or k + = k = k = nn ( + ) = 6 n + n 6= ( n + n = ( n+ ( n = n+ = or n = n= n=
28 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual p( p = 8 p p 8= ( p p 6) = ( p ( p+ ) = p = or p+ = p= p= x(x+ = x + x+ x + 5x = x + x+ x = x = ( zz ) z= z + z 6z z = z + 7z = z = 7 7q = 9q 7q 9q= 9 q(q = 9q= or q = q= q= w = w w w= 7 w(w ) = 7w= or w = w= w= ( c c) = ( cc ) = c= or c = c= c= ( d + d) = d(d + = d = or d + = d = d = y y y+ = y ( y ( y = ( y ( y+ )( y ) = y = or y+ = or y = y = y = y = t + t 6t = t ( t+ ) 6( t+ ) = ( t+ )( t+ ( t = t+ = or t+ = or t = t = t = t = ( x ( x+ ) = 8 x + x 8= x + x = ( x+ ( x = x+ 5= or x = x= 5 x= ( w+ ( w = w + w 5 = w + w 5= ( w+ 7)( w = w+ 7= or w 5= w= 7 w= 5 ( p+ )( p+ = p p + 5p+ 6= p p + 6p+ 5= ( p+ ( p+ = p+ 5= or p+ = p= 5 p=
29 Chapter 6 Factoring Polynomials 7. ( k 6)( k = k k 7k + 6= k k 6k + 8= ( k ( k ) = k = or k = k = k = 6. (a) 6a 7a = (a+ (a = a+ = or a = a= a= 6a 7a = (a+ (a Problem Recognition Exercises. (a) x + 6x 7 = ( x+ 7)( x ( x+ 7)( x =. (a) x+ 7= or x = x= 7 x= c + 8c+ = ( c+ 6)( c+ ) ( c+ 6)( c+ ) =. (a) c+ 6= or c+ = c= 6 c= y + 7y+ = (y+ ( y+ (y+ ( y+ = y+ = or y+ = y = y =. (a) x 8x+ 5 = (x ( x (x ( x = x 5= or x = 5 x= x= 5. (a) 5q + q = (5q ( q+ = 5q = or q+ = q= 5 q= 5q + q = (5q ( q+ 7. (a) 8. (a) 9. (a). (a). (a) a 6 = ( a+ 8)( a 8) = a+ 8= or a 8= a= 8 a= 8 a 6 = ( a+ 8)( a 8) v = ( v+ ( v = v+ = or v = v= v= v = ( v+ ( v b 8 = (b+ 9)(b 9) (b+ 9)(b 9) = b+ 9= or b 9= 9 9 b= b= 6t 9 = (6t+ 7)(6t 7) (6t+ 7)(6t 7) = 6t+ 7= or 6t 7= 7 7 t = t = 6 6 8x + 6x+ 6= (x + 8x+ = (x+ (x+ = x+ = or x+ = x= x= 8x + 6x+ 6= (x+ (x+
30 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. (a) y + y+ = (y + y+ 8) = (y+ ( y+ ) = y+ = or y+ = y= y= y + y+ = (y+ ( y+ ). (a) x 8x x= x( x 8x ) = xx ( ( x+ ) xx ( ( x+ 8) = x= or x = or x+ = x= x= x=. (a) k + 5k k = k( k + 5k = kk ( + 7)( k ) kk ( + 7)( k ) = k = or k + 7= or k = k = k = 7 k = 5. (a) b + b 9b 9= b ( b+ 9( b+ = ( b+ ( b 9) = ( b+ ( b+ ( b = b+ = or b+ = or b = b= b= b= b + b 9b 9 = ( b+ ( b ( b+ 6. (a) x 8x x+ = x ( x 8) ( x 8) = ( x 8)( x = ( x 8)( x+ )( x ) = x 8 = or x+ = or x = x= 8 x= x= x 8x x+ = ( x 8)( x+ )( x ) 7. s 6s+ rs r = s( s + r( s = ( s ( s+ r) 8. 6t + t+ tu+ 5u = t(t+ + 5 u(t+ = (t+ (t+ 5 u) 9. 8x x= x(x = x(x (x+ = x= or x = or x+ = x= x= x =. b 5b= b( b = bb ( ( b+ = b= or b 5 = or b+ 5 = b= b= 5 x= 5 x x + x= x( x x+ = xx ( = x= or x = or x = x= x= x=. t + 8t + 7t = t( t + 6t+ 9) = ( tt+ = t = or t+ = or t+ = t = t = t =.. 7c c+ = 7( c + c) 7c c+ = 7c + 7c = 9c c = 9 c =. ( z z+ = 7+ 6z 6z + z = 7+ 6z z = 7 7 z =
31 Chapter 6 Factoring Polynomials 5. 8w + 7 = ( w) + ( = (w+ (w 6w+ 9) q = ( q) ( = (q ( q + q+ 5z + z = 7 5z + z 7= (5z+ 7)( z = 5z+ 7= or z = 7 z = 5 z = h + 5h= 6 h + 5h+ 6= (h+ ( h+ 6) = h+ = or h+ 6= h= h= 6 bb ( + 6) = b b + 8b= b b + 7b+ = (b+ )( b+ = b+ = or b+ 5= b= b= 5 y + = y( y y + = y y y + y+ = (y+ ( y+ = y+ = or y+ = y = y =. 5(x (x+ = x x 5 6x = x x 7= x 7x = x =. 6a= (a 6a= 8a+ 6a= 8a+ a = a = s = 6 s 6= ( s 6) = ( s+ ( s = s+ = or s = s= s= 8v = 6 8v 6 = 9(9v = 9(v+ )(v ) = v+ = or v = v= v= ( x ( x = 6 x 7x+ = 6 x 7x+ 6= ( x 6)( x = x 6= or x = x= 6 x= ( x+ ( x+ 9) = x + x+ 5 = x + x+ = ( x+ ( x+ ) = x+ = or x+ = x= x= Section 6.8 Practice Exercises. (a) x + x + (c) x +
32 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual (d) LW (e) (f) bh a + b = c. (6x+ ( x+ = 6x+ = or x+ = x= 6 x=. 9 x(x+ ) = 9x= or x+ = x= x= x = (x+ (x = x+ = or x = x= x= x 5x= 6 x 5x 6= ( x 6)( x+ = x 6= or x+ = x= 6 x= xx ( ) = x x+ = ( x = x = or x = x= x= 6x 7x = (6x+ ( x ) = 6x+ 5= or x = 5 x= 6 x= 8. A factored expression must be set equal to zero to use the zero product rule. 9. Let x = the number. Then, x + = 6 x 9 = ( x+ 7)( x 7) = x+ 7= or x 7= x= 7 x= 7 The numbers are 7 and 7.. Let x = the number. Then, x+ x = 6 x + x 6= (x+ 9)( x = x+ 9= or x = 9 x= x= 9 The numbers are and.. Let x = a number. Then, + 6x= x 8 x 6x = ( x+ ( x = x+ = or x = x= x= The numbers are and.. Let x = a number. Then, x = x+ x x = ( x+ ( x = x+ = or x 5= x= x= 5 The numbers are and 5.. Let x = first integer. Then the next consecutive odd integer is (x + ). xx ( + ) = 6 x + x 6= ( x+ 9)( x 7) = x+ 9= or x 7= x= 9 x= 7 The numbers are 9 and 7, or 7 and 9.
33 Chapter 6 Factoring Polynomials 5. Let x = first integer. Then the next consecutive even integer is (x + ). xx ( + ) = 8 x + x 8= ( x+ 8)( x 6) = x+ 8= or x 6= x= 8 x= 6 The numbers are 8 and 6, or 6 and Let x = first integer. Then the next consecutive integer is (x +. x + ( x+ = 6 x + x + x+ = 6 x + x 6= ( x+ 6)( x = x+ 6= or x 5= x= 6 x= 5 The numbers are 6 and 5, or 5 and Let x = first integer. Then the next consecutive integer is (x + ). x + ( x+ ) = 5 x + x + x+ = 5 x + x 8= ( x + x = ( x+ 6)( x = x+ 6= or x = x= 6 x= The numbers are 6 and , or and Let x = width of the painting and x + is the length of the painting. Then, A = (length)(width) 99 = xx ( + ) 99 = x + x x + x 99= ( x+ ( x ) = x = or x = 9 The height of the painting is ft and the width is 9 ft. 8. Let x = length of painting and x be the length of the painting. Then, A = (length)(width) = xx ( ) = x x x x = ( x+ ( x = x = or x = The painting has length in. and width in. 9. Let x = length and x be the width. (a) A= lw 8 = xx ( x x 8= ( x+ ( x 7) = x = or x = 7 The dimensions are 7 m by m. P = w + l P = ( + (7) P = 8 + = m. Let x = length and x 7 be the width. (a) A= lw 78 = xx ( 7) x 7x 78= ( x+ 6)( x = x = 6 or x = The dimensions are in. by 6 in. P = w + l P = (6) + ( P = + 6 = 8 in.. Let x = height and x + = the base. A= bh = ( x)( x+ 8 = x + x x + x 8= ( x+ 7)( x = x+ 7= or x = x = 7 x = The height is ft and the base is 7 ft.
34 6 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. Let x = base and x + 5 = the height. A= bh 5 = ( x)( x+ 5 = x + 5x x + 5x 5 = ( x+ ( x = x+ 5 = or x = x = 5 x = The base is cm and the height is 5 cm.. Let x = base and x 7 = the height. A= bh = ( x)(x 7) = x 7x x 7x = (x+ 8)( x = x+ 8= or x 5= 8 x = x = 5 x 7= ( 7 = 8 The base is 5 cm and the height is 8 cm.. Let x = height and x = the base. A= bh 6 = ( x)(x ) = x x x x = (x x 6) = (x+ ( x ) = x+ = or x = x = x = x = () = 6 The base is 6 ft and the height is ft. 5. If you let h =, then, = 6t + = 6( t 9) 6( t+ ( t = t+ = or t = t = t = It will take seconds to hit the ground. 6. If you let h =, then = 6t + 56 = 6( t 6) 6( t+ ( t = t+ = or t = t = or t = It will take seconds to hit the ground. 7. Ground level is when h =. Then, = 6t + t = 8 t(t 8t = or t = t = t = =.5 The times are seconds and.5 seconds. 8. Ground level is when h =. Then, = 6t + 6t = 6 tt ( 6t = or t = t = t = The times are seconds and seconds. 9. Pictures may vary.. Given a right triangle with legs a and b and hypotenuse c, then a + b = c.
35 Chapter 6 Factoring Polynomials c = a + b c c = + 7 = 65 c 65 = ( c+ ( c = c = 5 or c = 5 c = 5 cm c = a + b c c = + = 5 c 5 = ( c+ ( c = c = 5 or c = 5 c = 5 m c = a + b 7 = a = a + 6 a 5 = ( a+ ( a = a = 5 or a = 5 a = 5 in. c = a + b 5 = 9 + b 5 = 8+ b b = ( b+ ( b = b = or b = b = yd 5. Let x = c. c = a + b x x = 6 + = x = ( x+ )( x ) = x = or x = The brace is in. long. 6. Let h = a. c = a + b 5 = h + h = 8 h 8 = ( h+ 9)( h 9) = h = 9 or h = 9 The height is 9 km 7. Let h = height of the kite. Then, a = h. c = a + b = ( h + 9 = h h h h = ( h+ 7)( h 9) = h = 7 or h = 9 The kit is 9 yd high. 8. Let c = distance between the two cars. Then, a = 8 and b = 6. c = a + b c = c = 6 c 6 = ( c+ 8)( c 8) = c = 8 or c = 8 They were 8 mi apart. 9. Let x = distance between the base of the ladder and the bottom of the house. Then, the distance between the top of the ladder and the ground is x + 7 ft. c = a + b 7 = x + ( x+ 7) 89 = x + x + x+ 9 x + x = ( x+ ( x 8) = x = 5 or x = 8 The bottom of the ladder is 8 ft from the house. The distance from the top of the ladder to the ground is 5 ft.
36 8 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. Let x = distance traveled by the first boat. Then, the distance traveled by the second boat is x + mi. c = a + b 5 = x + ( x+ 5 = x + x + x+ x + x = ( x+ ( x = x = or x = The first boat traveled mi; the second boat traveled mi.. Let x = hypotenuse. Then, one leg is x m and the other leg is x m. x = ( x + ( x ) x = x 8x+ 6+ x x+ x = x x+ x x+ = ( x ( x ) = x= or x= The hypotenuse is m.. Let x = length of the shorter leg. Then, the longer leg is x cm and the hypotenuse is x + cm. (x+ = x + (x x + x+ = x + x x+ x + x+ = 5x x+ x 8x= xx ( 8) = x= 8 or x= The length of the shorter leg is 8 cm. Group Activity. Answers will vary. For example: x x + x= x( x 5x+. Answers will vary. For example: xy + 5y x = (x + (5y 7). Answers will vary. For example: x 5x = ( x+ )( x 7). Answers will vary. For example: 6x 7x 5 = (x+ (x 5. Answers will vary. For example: 7x x x= 7 x( x ( x+ 6. Answers will vary. For example: 6x 8 y = (x + 9 y)(x 9 y) 7. Answers will vary. For example: 6 5 b = (6 5 b)(6 + b+ 5 b ) 8. Answers will vary. For example: u + 7 v = ( u+ v)( u uv+ 9 v ) 9. Answers will vary. For example: ( x ( x+ 7) = x + x 8=. Answers will vary. For example: x x+ = x + 6x= Chapter 6 Review Exercises Section 6.. GCF: a b. GCF: x + 5. GCF: c(c. GCF: yz or yz x + x 8x= x(x+ x 5 w y w y = w y ( w y ) t + 5 t = t( t or t( t + 6 u u = u(6u+ or u( 6u 9. b(b + ) 7(b + ) = (b + )(b 7)
37 Chapter 6 Factoring Polynomials 9. (5x + 9) + 8x(5x + 9) = (5x + 9)( + x).... 7w + w+ wb+ b= 7 w( w+ ) + b( w+ ) = ( w+ )(7 w+ b) b b+ yb y = b( b ) + y( b ) = ( b )( b+ y) Section y 5y y+ 9 = (y 5y y+ = (5 y( y ( y ) = (y (5 y 6a a ab+ a b= a(6 a b+ ab) = a(( a) b( a)) = a( a)( b) x x+ = x 7x x+ = xx ( 7) ( x 7) = ( x ( x 7) 6. y 9y+ 88 = y y 8y+ 88 = yy ( 8( y = ( y 8)( y 7. 6z+ z 7= z 6z 7 = z + 6z z 7 = zz ( + 6) ( z+ 6) = ( z ( z+ 6). m + 6m + 8m = m ( m + m+ = m ( m + 5m+ 8m+ = m ( m( m+ + 8( m+ ) = m ( m+ 8)( m+.. t + t 6 = ( t t+ 6) = ( t t 8t+ 6) = (( tt ) 8( t )) = ( t 8)( t ) w w+ = ( w + w ) = ( w + 5w w ) = ( ww ( + ( w+ ) = ( w ( w+ a + ab+ b = a + ab+ ab+ b = aa ( + b) + ba ( + b) = ( a+ b)( a+ b). c cd 8d = c + cd 6cd 8d = cc ( + d) 6 dc ( + d) = ( c 6 d)( c+ d). Section Different 6. Both negative 7. Both positive q q= q q+ 9 p w+ 6pw+ 6w = wp ( + p+ ) = q + q q+ 9 = qq ( + ( q+ = ( q ( q+ = wp ( + p+ p+ ) = wpp ( ( + ) + ( p+ ) = wp ( + ( p+ ) 8. Different 9... y 5y = y 8y+ y = yy ( + ( y = (y+ ( y w 5w 6= w 8w+ w 6 = ww ( ) + ( w ) = (w+ ( w ) z + 9z+ = z + z+ 5z+ = (5 z z+ ) + 5(5z+ ) = (z+ (5z+ )
38 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual z + 6z 9= 8z + z 6z 9 = z(z+ (z+ = (z (z+ p 5p+ is prime. 5r r+ 7 is prime. w 6w 7 = ( w 6w 7) = ( w 9w+ w 7) = ( ww ( 9) + ( w 9) = ( w+ ( w 9) Section 6.. 5,., c 5c = c 6c+ c = cc ( ) + ( c ) = ( c )(c+ y + y+ = y + y+ y+ = yy ( + + ( y+ = ( y+ (y y 8y 8 = ( y 6y 6) 9c cd + 5d = ( y 8y+ y 6) = ( yy ( 8) + ( y 8)) = ( y+ )( y 8) = 9c 5cd 5cd + 5d = ( c c 5 d) 5 d(c 5 d) = (c 5 d)(c 5 d) = (c 5 d) x + x+ 6 = x + 6x+ 6x+ 6 = xx ( + 6) + 6( x+ 6) = ( x+ 6)( x+ 6) = ( x + 6) t + tw+ w = t + tw+ tw+ w = tt ( + w) + ( t+ w) = ( t+ w)( t+ w) x + 7x 5= x + x x = x ( x + ( x + = ( x + (x w + 7w + = w + 5w + w + = w ( w + + ( w + ) = ( w + ( w + ) p 8pq+ 5q = p pq 5pq+ 5q = p( p q) 5 q( p q) = ( p q)( p 5 q) g + 7gh + h = (g + h)( g + h) m mn+ 5 n = (6 m n)(m 5 n) v v = ( v ) v = ( v ) + v v = v ( v + ( v + = ( v ( v v + v 6 = (v + v = (v + 5v v = (5 v(v+ (v+ ) = (v+ (5v s + s = (s + s = (s + 8s 5s = ( ss ( + ) 5( s+ )) = ( s+ )(s. x + 7x + = ( x ) + 7x + = ( x ) + x + 5x + = x ( x + ) + 5( x + ) = ( x + ( x + )
39 Chapter 6 Factoring Polynomials ab ab + ab = ab( a ab + b ) = ab( a 6ab ab + b ) = ab( a( a 6 b) b( a 6 b)) = ab( a 6 b)( a b) 6 5 z + 8z z = z ( z + z m+ 9m prime + 6p + 9p prime = z ( z + 7z z = z ( z( z+ 7) ( z+ 7)) = z ( z+ 7)( z y + ; this is a sum of squares, not a difference. y + y+ 6 = ( y+ 6) t + 6t+ 6 = ( t+ 8) 9a a+ = (a ) 5x x+ 6 = (5x v v = ( v + v+ = ( v + ) x + x 5= ( x x+ = ( x 57. 9x + x+ = (7 x) + (7 x)( + = (7x c 8= ( c 9) = ( c + ( c 7x y = (6 x y ) = (6 x + y)(6 x y) 58. Section w 6wz+ z = ( w) + ( w)( z) + ( z) = ( w z) a b = ( a b)( a+ b) 6. a + b is prime p + p 6p 8 = p + p 6( p+ = p ( p+ 6( p+ = ( p 6)( p+ = ( p ( p+ ( p+ k 8 k + k = ( k )( k)( + k) or = ( k ) ( k + ) a 9 = a 7 = ( a+ 7)( a 7) d 6 = d 8 = ( d + 8)( d 8) 8t = (9 t) = ( + 9 t)( 9 t) Section a + b = ( a+ b)( a ab+ b ) 78. a b = ( a b)( a + ab+ b ) k = (5 k) = (+ 5 k)( 5 k) x + 6; this is a sum of squares, not a difference a = + a = ( + a)( a+ a ) = ( + a)(6 a+ a )
40 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual 8. 5 b = 5 b p q = (5 b)(5 + 5 b+ b ) = (5 b)(5 + 5 b+ b ) + 8 = ( p ) + = ( p + )(( p ) p + ) = ( p + )( p p + = ( q ) 7 = q ( q ) + q + = q q + q x 8= 6( x 8) = 6( x ) = 6( x )( x + x+ 8. 7y + 7= 7( y = 7( y+ ( y y+ x 6 x= xx ( 6) = xx ( + 6)( x 6) q 6 q= q( q ) = qq ( ( q + q+ 6) 8h + = (h + m 8 m= m( m 8) 9. Section 6.7 m = ( m = ( m ( m + m+ 9. The equation (x (x + = can be solved directly by the zero product rule because it is a product of factors set equal to zero. 9. (x (x + ) = x = or x+ = x= x= 95. (a 9)(a = a 9= or a = a= 9 a= 96. w(w + (5w + ) = w= or w+ = or 5w+ = w= w= w= u(u 7)(u 9) = 6u = or u 7 = or u 9 = u = u = 7 9 u = 98. 7k 9k = (7k + ( k ) = 7k + 5= or k = 5 k = 7 k = x + x x = x ( x+ ( x+ = ( x+ ( x = ( x+ ( x+ ( x 5pq q = 5 qp ( q) = 5 q( p + q)( p q) 8 n+ n = n(8 + n ) = n( + n)( n+ n ) 99.. h h 6= (h+ ( h 6) = h+ = or h 6= h= h= 6 q = ( q+ ( q = q+ = or q = q= q=
41 Chapter 6 Factoring Polynomials r = 5 r 5 = ( r+ ( r = r+ 5= or r 5= r = 5 r = 5 5v v= v(5v = v= or 5v = v= v= 5 xx ( 6) = 8 x 6x+ 8= ( x ( x ) = x = or x = x= x= 6t + 6t = 5 6t + 6t+ 5 = (6t+ (6t+ = 6t + 5= 5 t = 6 9s + s= 9s + s+ = (s+ )(s+ ) = s + = s = ( y + = y y + = y y y+ = (y )( y 6) = y = or y 6= y = y = Section 6.8 ( p 66) = p p + p = ( p ( p+ = p = or p+ = p= p= y 8y = 8y y 8y + 8y = yy ( 9y+ = yy ( 7)( y ) = y = or y 7 = or y = y = or y = 7 or y = x x= xx ( = xx ( + )( x ) = x= or x+ = or x = x= or x= or x=. Let x = height and x + be the base. A = (height)(base) 78 = ( x)(x+ 78 = x + x x + x 78= (x+ ( x 6) = x= or x= 6 The height is 6 ft and the base is ft.. At h =, the ball is ground level. = 6x + 6x = 6 xx ( x = or x = The ball is at ground level at and second.
42 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. Let x represent the length of the ramp. + 5 = x + 5 = x 69 = x = x The ramp is ft long.. Let x = shorter leg. Then, the other leg is x + and the hypotenuse is x. x + ( x+ ) = (x ) x + x + x+ = x 8x+ x x= xx ( 6) = x = or x = 6 The legs are 6 ft and 8 ft; the hypotenuse is ft. Chapter 6 Test x x+ 6x = x(5x + x ) 7a 5 a + 5a= 7( a a( a = ( a (7 a) 6w w+ 7 = (6w ( w 7) 69 p = p = ( + p)( p) q 6q+ 6 = ( q 8) 8 + t = ( + t)( t+ t ). Let x = a number. Then, 6 x = x 6 = ( x+ 8)( x 8) = x = 8 or x = 8 The numbers are 8 and x = first integer and x + is the next integer. xx ( + = ( x+ x+ + x + x= 8x+ + x 7x 58 = ( x+ )( x 9) = x = and x = 9 The numbers are and or 9 and a + a+ = ( a+ ( a+ 8) x + x = ( x+ 7)( x 6) y 7y+ 8 = (y ( y 8) 6z + 9z+ 8 = (z+ (z+ 8) 9t = (t+ (t v 8 = ( v+ 9)( v 9) a + 7ab+ 5b = ( a + 9ab+ 8 b ) = ( a+ 6 b)( a+ b) 6. Let x = height, x + be the base. A= bh 8 = x(x+ 6 = x + x x + x 6= (x+ 9)( x = 9 x= or x= The height is m and the base is 9 m.. c = ( c + ( c = ( c + ( c+ ( c 5. xy 7x + y = x( y 7) + ( y 7) = ( y 7)( x p is prime. u + u = ( u u+ ) = ( u )( u t 75 = (t = (t+ (t
43 Chapter 6 Factoring Polynomials y 5y+ 5 = 5( y y+ = 5( y ( y = 5( y q + q= 7 q(q+ ) x + x 8x = x (x+ (x+ = (x+ ( x = (x+ ( x+ )( x ) y 5 = ( y ( y + 5y+ mn 8 = ( mn+ 9)( mn 9).. x(5x+ = 5x + x = (5x ( x+ = 5x = or x+ = x= 5 x= y + y 9y 9 = y ( y+ 9( y+ = ( y 9)( y+ = ( y+ ( y ( y+ = y+ = or y = or y+ = y = or y = or y = a 6b = 6( a b ) = 6( a+ b)( a b) 6 6x 7y = (x y )(6 x + xy + 9 y ) x y 6xy y= y( x ( x+ ) 7. (x ( x+ = x = or x+ 5= x= x= x 7x= xx ( 7) = x= or x 7= x= x= 7 x 6x= 6 x 6x 6= ( x+ )( x 8) = x+ = or x 8= x= x= 8. Let x = width and x + is the length. A= lw = (x + )( x) = x + x x + x = ( x + x 56) = ( x+ ( x = x = or x = The tennis court is yd by 6 yd.. Let x represent the first odd integer. Then x + represents the second odd integer. xx ( + ) = 5 x + x= 5 x + x 5= ( x+ 7)( x = x+ 7= or x 5= x= 7 or x= 5 x+ = 5 or x+ = 7 The two integers are 5 and 7 or 5 and 7.. Let x represent the length of the base. Then x 5 represents the height of the triangle.
44 6 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual A = (base)(height) = ( x)( x 8 = ( x)( x 8 = x 5x x 5x 8= ( x ( x+ 7) = x = or x+ 7 = x= or x= 7 x 5= 7 The base is in. and the height is 7 in. 5. Let x = shorter leg, x the longer leg, and x the length of the hypotenuse. ( x) + (x = (x ) x + 9x 8x+ 9= 9x x+ x 6x+ 5= ( x ( x = x = or x = 5 The shorter leg is 5 ft. 6. h= 6t + 6 = 6t + 6 = 6( t = ( t+ )( t ) t+ = or t = t = t = The stone hits the ground in sec. Cumulative Review Exercises Chapters 6. 5 ( ( = ( = = 7 = 5. 5 ( t+ = t+ 5 t 8= t+ t = t+ 5t = 5 5 t = = 5. x y = 8 y = 8 x y 8 x = x 8 8 x y = or y =. Let x = number of quarters, (x + ) the number of nickels, and (x the number of dimes. Value Value Value of + of + of = $.8 quarters nickels dimes.5x+.5( x+ ) +.( x =.8.5x+.5x+. +.x. =.8 5..x = x = = There are quarters, nickels, and 7 dimes. 5 5 x 5 5 x 5 5 x [, ) 6. (a) Yes m = (c) (, (d) = x + x = (, ).
45 Chapter 6 Factoring Polynomials 7 (e).. (p 5p (p = 8p p p p + 5 p+ = 8p p + p+ (w 7) = ( w) ( w)(7) + 7 = w 8w (a) Vertical line Undefined (c) (5, ) (d) Does not exist y y = m( x x ) y 5= [ x ( ] y 5= ( x+ y 5= x+ 9 y = x x y = 5x 6y = Multiply the first equation by to obtain opposite coefficients on y. Then add the equations and solve the resulting equation for x. x+ 6y = 8 5x 6y = x = 5 ( y = y = y = 6 y = Solution: (5, ). y y 7 y + y + 5y = y y y y 5y 7 = y 5y r + 5r + 5r+ r r + r + r 5r + ( r r ) 5r + r (5r 5 r ) 5r 5r (5r 5 r) r + (r r + 5r + 5r+ + r 5 7 c c c = = c = c c c 7 6ab 5 9 = a b = a b = 5 9 ab 8 8 = 6 = w 6 = ( w ) ( 6) = ( w + ( w = ( w + ( w+ )( w ) 8. ax + bx ya 5yb = ( x a+ b ( y a+ b = ( a+ 5 b)(x y) 9. x 8x 5 = (x+ (x ab
46 8 Miller // O Neill // Hyde Beginning Algebra Instructor s Solution Manual. x(x (x + = x= or x = or x+ 5= x= x= x= 5
Algebra I. Book 2. Powered by...
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