Chapter 11
Exercises page 128
1a- x2 + 8x + 16 = (x + 4)2
b- x2 – 4x + 4 = (x – 2)2
c- (2x + 2)2 = 4x2 + 8x + 4
d- (3 + 2x)2 = 9 + 12x + 4x2
e- (x – 5)2 = x2 – 10x + 25
f- (x + 6)2 = x2 + 12x + 36
g- x2 – 16 = (x – 4)(x + 4)
h- 9x2 – 25 = (3x -5)(3x + 5)
i- x2 – 2 = (x-√2)(x + √2)
j- 16x2 – 7 = (4x - √7)(4x + √7)
k- 3x2 – 49 = (√3x – 7)(√3x + 7)
l- 100 – 3x2 = (10 - √3x)(10 + √3x)
2a- x2 -36 = (x – 6)(x + 6)
b- 16x2 – 1 = (4x – 1)(4x + 1)
c- x2 – 25 = (x – 5)(x + 5)
d- 9 – x2 = (3 – x)(3 + x)
e- 9 – 4x2 = (3 – 2x)(3 + 2x)
f- 3 – 4x2 = (√3 – 2x)(√3 + 2x)
g- (x – 1)2 – 4 = [(x – 1) – 2] [(x – 1) + 2] = (x – 3)(x + 1)
h- 4x2 – (3x + 2)2 = [2x – (3x + 2)] [2x + (3x + 2)] = (- x - 2)(5x + 2)
3a- 4x5 + 5x4 – 6x2 + 8x3 + x2 – 4x4 – 4x5
= 4x5 – 4x5 + 5x4 – 4x4 + 8x3 – 6x2 + x2
= x4 + 8x3 – 5x2
Degree = 4
b- x3 – 2x2 + x/3 + 3x2 – 5/2x3 + 1 – 2/3x
= x3 – 5/2x3 – 2x2 + 3x2 + 1/3x – 2/3x + 1
= -3/2x3 + x2 – 1/3x + 1
Degree = 3
c- - x7 + 2x5 – x4 + 3x2 +5x4 + 2x6 – x + x7 + 12
= - x7 + x7 + 2x6 + 2x5 – x4 + 5x4 + 3x2 – x + 12
= 2x6 + 2x5 + 4x4 + 3x2 – x + 12
Degree = 6
d- 3x4 + x5 + x6 + 2x5 – x6 – 3x5 + 4
= x6 – x6 + x5 + 2x5 – 3x5 + 3x4 + 4
= 3x4 + 4
Degree = 4
4a- x(2x – 1) + (x + 1)(1 -3x) = 2x2 – x + x – 3x2 + 1 – 3x
= - x2 – 3x + 1
b- - 4(3 – 2y) + 5y(y – 2) = - 12 + 8y + 5y2 – 10y
= 5y2 – 2y – 12
c- z2(z2- z + 1) – (z2 + 1)(z – 1) = z4 – z3 + z2 – z3 + z2 – z + 1
= z4 – 2z3 + 2z2 – z + 1
d- (u2 + u + 1)(u + 1) – 2u(1 – u)
= u3 + u2 + u2 + u + u + 1 – 2u + 2u2
= u3 + 4u2 + 1
e- 2v(v3 – v2 + v – 1) + (v – 2)(v2 – v – 1)
= 2v4 – 2v + 2v2 – 2v + v3 – v2 – v – 2v2 + 2v + 2
= 2v4 – v3 – v2 – v + 2
f- (t3 – 2t + 1)(t4 – 1) + t3(t3 – t2 + 1)
= t7 – t3 – 2t6 + 2t + t4 – 1 + t6 – t5 + t3
= t7 – t6 – t5 + t4 + 2t – 1
g- 4(3a3 – a – 1) + 2a/3(3 – a) = 12a3 – 4a – 4 + 2a – 2/3a2
= 12a3 – 2/3a2 - 2a – 4
h- (b + 5)(b2 + 1) – b(b2 – b + 3) = b3 + b + 5b2 + 5 – b3 + b2 – 3b
= 6b2 - 2b + 5
i- c2 + 9c – 2(c2 – 1)2 = c2 + 9c – 2(c4 – 2c2 + 1)
= c2 + 9c – 2c4 + 4c2 – 2
= -2c4 + 5c2 + 9c – 2
j- (p – 1)(p + 1)2 + (2p – 5)(11p2 – 1)
= (p – 1)(p2 + 2p + 1) + (22p3 – 2p – 55p2 + 5)
= p3 + 2p2 + p – p2 – 2p – 1 + 22p3 – 2p – 55p2 + 5
= 23p3 – 54p2 – 3p + 4
k- (q3 + 1)2 + (2q – 3)(q3 – 1) + 4
= q6 + 2q3 + 1 + 2q4 – 2q – 3q3 + 3 + 4
= q6 + 2q4 – q3 – 2q + 8
5a- p(x) + q(x) = (3x2 – 3x – 4) + (3x2 – 2x + 1)
= 6x2 – 5x – 3
p(x) – q(x) = (3x2 – 3x – 4) – (3x2 – 2x + 1)
=-x–5
p(x).q(x) = (3x2 – 3x – 4)(3x2 – 2x + 1)
= 9x4 – 6x3 + 3x2 – 9x3 + 6x2 – 3x – 12x2 + 8x – 4
= 9x4 – 15x3 – 3x2 + 5x – 4
b- p(x) + q(x) = (2x3 – x2 + 1) + (-2x3 + x)
= - x2 + x + 1
p(x) – q(x) = (2x3 – x2 + 1) - (-2x3 + x)
= 4x3 – x2 – x + 1
p(x).q(x) = (2x3 – x2 + 1)(-2x3 + x)
= -4x6 + 2x4 + 2x5 – x3 – 2x3 + x
= -4x6 + 2x5 + 2x4 – 3x3 + x
c- p(x) + q(x) = (x3 – x + 2) + (x2 + 2x + 4)
= x3 + x2 + x + 6
p(x) – q(x) = (x3 – x + 2) - (x2 + 2x + 4)
= x3 – x2 – 3x – 2
p(x).q(x) = (x3 – x + 2)(x2 + 2x + 4)
= x5 + 2x4 + 4x3 – x3 – 2x2 – 4x + 2x2 + 4x + 8
= x5 + 2x4 + 3x3 + 8
d- p(x) + q(x) = (4x2 + x) + (-4x2 + 3)
=x+3
p(x) – q(x) = (4x2 + x) - (-4x2 + 3)
= 8x2 + x – 3
p(x).q(x) = (4x2 + x)(-4x2 + 3)
= -16x4 + 12x2 – 4x3 + 3x
= -16x4 – 4x3 + 12x2 + 3x
e- p(x) + q(x) = (2x5 – 1) + (2x3 + 7)
= 2x5 + 2x3 + 6
p(x) – q(x) = (2x5 – 1) - (2x3 + 7)
= 2x5 – 2x3 – 8
p(x).q(x) = (2x5 – 1)(2x3 + 7)
= 4x8 + 14x5 – 2x3 – 7
6a- p(1/2) = 2(1/2)3 – 4(1/2)2 + 1/2 + 1
= 2(1/8) – 4(1/4) + 1/2 + 1
= 1/4 – 1 + 1/2 + 1
= 3/4
b- p(1) = 12 – 3(1) + 2 = 1 – 3 + 2 = 0
c- p(-2) = 17(-2)5 – 14(-2)4 + 2(-2) + 6
= -544 – 224 – 4 + 6
= -766
d- p(0.1) = (0.1)2 + 0.1 + 4 = 0.01 + 0.1 + 4 = 4.11
e- p(√2) = 2(√2)2 + (√2/3)2 - √2
= 2(4) + 2/3 - √2 = 26/3 - √2
f- p(√2) = (1 -√2)2(√2)2 = (2 + √2)(√2) – 6
= 2 - 2√2 + 2√2 + 2 – 6
= -2
7a- p(x) = 2x2 – 7x + 5
p(a) = p(1) = 2(1)2 – 7(1) + 5 = 2 – 7 + 5 = 0
p(b) = p(2) = 2(2)2 – 7(2) + 5 = 8 – 14 + 5 = -1
p(c) = p(2.5) = 2(2.5)2 – 7(2.5) + 5 = 12.5 – 17.5 + 5 = 0
so, a and c are roots of p(x)
b- p(x) = -2x3 – x2 + 7x + 6
p(a) = p(2) = -2(2)3 – 22 + 7(2) + 6 = -16 – 4 + 14 + 6 = 0
p(b) = p(-1) = -2(-1)3 – (-1)2 + 7(-1) + 6 = 2 – 1 – 7 + 6= 0
p(c) = p(1) = -2(1)3 – 12 + 7(1) + 6 = -2 – 1 + 7 + 6 = 10
so, a and b are roots of p(x)
c- p(x) = x2 + 4x + 3
p(a) = p(-1) = (-1)2 + 4(-1) + 3 = 1 – 4 + 3 = 0
p(b) = p(1) = (1)2 + 4(1) + 3 = 1 + 4 + 3 = 8
p(c) = p(-3) = (-3)2 + 4(-3) + 3 = 9 – 12 + 3 = 0
so, a and c are roots of p(x)
d- p(x) = x2/2 – 7x/2 – 4
p(a) = p(-1) = (-1)2/2 – 7(-1)/2 – 4 = 1/2 + 7/2 – 4 = 4 – 4 = 0
p(b) = p(8) = (8)2/2 – 7(8)/2 – 4 = 32 – 28 - 4 = 0
p(c) = p(1) = (1)2/2 – 7(1)/2 – 4 = 1/2 - 7/2 – 4 = -3 – 4 = -7
so, a and b are roots of p(x)
8a- p(r) = 0 → p(2) = 0 → a(2)2 + (a – 1)(2) + 1 = 0
4a + 2a – 2 + 1 = 0
6a – 1 = 0 → a = 1/6
b- p(3) = 0 → (2a – 1)(3)2 + 3a(3) + 2 = 0
18a – 9 + 9a + 2 = 0
27a = 7 → a = 27/7
c- p(-2) = 0 → (a2 + 1)(-2)2 + 2a(-2) + 4a – 4 = 0
4a2 + 4 – 4a + 4a – 4 = 0
4a2 = 0 → a = 0
d- p(1) = 0 → a(1)3 + (1 – a)(1)2 + 1 +a – 1 = 0
a+1–a+1+a–1=0
a + 1 = 0 → a = -1
9a- p(x) = 0 → (2a – 1)x2 + (3 – b)x + 1/2(2 – 5c) = 0
2a - 1 = 0 and 3 – b = 0 and 2 – 5c = 0
a = 1/2 and b = 3 and c = 2/5
b- p(x) = 0 → (a2 - 1)x2 + 2bx + c = 0
a2 – 1 = 0 and 2b = 0 and 1 – c = 0
a = 1 or a = -1 and b = 0 and c = 1
c- p(x) = 0 → (a + √3)x2 + (b - 4)x + c = 0
a + √3= 0 and b - 4 = 0 and c = 0
a = - √3 and b = 4
d- p(x) = 0 → (0.35 – 5a)x3 + (2b - 1)x2 + (c2 – 4)x + c + 2 = 0
0.35 – 5a = 0 and 2b - 1 = 0 and c2 - 4 = 0 and c+ 2 = 0
a = 0.07 and b=1/2 and c = 2 (rej) or c = -2 and c = -2
e- p(x) = 0 → (a2+ 2a)x2 + (2a + b)x + 3c - 7 = 0
a2 + 2a = 0 and 2a + b = 0 and 3c - 7 = 0
a = 0 or a = -2 and b = 0 or b = 4 and c = 7/3
10a- p(x) = q(x)
x3 + (2b – 1)x2 – 3x – 2 = (a – 1)x3 + 3x2 – 3x + 2c + 1
 1=a–1 → a=2
 2b – 1 = 3 → b = 2
 -2 = 2c + 1 → c = -3/2
b- p(x) = q(x)
(2a – 1)x3 + 4x2 + 5 = (3a - 5)x3 + (b – 1)x2 + (2c + 1)x + 5
 2a - 1 = 3a – 5 → a = 5 – 1 = 4
 4=b-1 → b=5
 0 = 2c + 1 → c = -1/2
c- p(x) = q(x)
(a + 5)x3 + 2x2 + x + 1 – b = 2x2 + (4c – 1)x + 4
 a+ 5 = 0 → a = -5
 1 = 4c - 1 → c = 1/2
 1 - b = 4 → b = -3
d- p(x) = q(x)
(a2 – 3)x2 + 2bx + 14 = x2 + (1 – b)x + 3c
 a2 – 3 = 1 → a2 = 4 → a = -2 or a = 2
 2b = 1 - b → b = 1/3
 14 = 3c → c = 14/3
e- p(x) = q(x)
(2a + b)x2 + 5x + (1 - √3)c = 3x2 + (a – 2b)x + 1 - √3
 2a + b = 3
a – 2b = 5 →x (-2) → -2a + 4b = -10
by elimination: 5b = -7 → b = -7/5
substitute b to get a: → a – 2(-7/5) = 5 → a = 11/5
 (1 - √3)c = 1 + √3
c = (1 + √3) / (1 - √3) → rationalize → c = (1 + √3)2 / 12 – (√3)2 = -2 - √3
11a- p(x) = 2x + 3 = 0
2x = -3 → x = -3/2
b- p(x) = x2 - 9 = 0
(x -3)(x + 3) = 0 → x = 3 or x = -3
c- p(x) = 4 – 2x = 0
2x = 4 → x = 2
d- p(x) = x2 – 2x + 1 = 0
(x – 1)2 = 0 → x = 1
e- p(x) = x(2x + 5) = 0
x = 0 or 2x + 5 = 0 → x = -5/2
f- p(x) = (x + 1)(x + 2) = 0
x + 1 = 0 or x + 2 = 0 → x = -1 or x = -2
12a- p(x) = x2 – 4x = x(x – 4)
p(x) = 0 → x(x – 4) = 0
x = 0 or x = 4
b- p(x) = x3 – 4x = x(x2 – 4) = x(x - 2)(x + 2)
p(x) = 0 → x(x - 2)(x + 2)= 0
x = 0 or x = 2 or x = -2
c- p(x) = x4 – 9x2 = x2(x2 – 9) = x2(x - 3)(x + 3)
p(x) = 0 → x2(x - 3)(x + 3) = 0
x = 0 or x = 3 or x = -3
d- p(x) = x3 + 2x2 + x = x(x2 + 2x + 1) = x(x + 1)2
p(x) = 0 → x(x + 1)2= 0
x = 0 or (x + 1)2 = 0 → x = -1
e- p(x) = (x + 1)2 – (3x + 5)2
= [(x + 1) – (3x + 5)] [(x + 1) + (3x + 5)]
= (-2x – 4)(4x + 6)
p(x) = 0 → (-2x – 4)(4x + 6) = 0
-2x – 4 = 0 or 4x + 6 = 0
x = -2 or x = -3/2
f- p(x) = 25x2 – (x + 1)2
= [(5x) – (x + 1)] [(5x) + (x + 1)]
= (4x – 1)(6x + 1)
p(x) = 0 → (4x – 1)(6x + 1)= 0
4x – 1 = 0 or 6x + 1 = 0
x = 1/4 or x = -1/6
13a- 1/(x + 1) is defined iff x + 1 ≠ 0
→x≠-1
→ x belongs to ] - ∞, -1[ U ]-1, +∞[
b- x/(x + 1) is defined iff x + 1 ≠ 0
→x≠-1
→ x belongs to ] - ∞, -1[ U ]-1, +∞[
c- (x + 1)/(2x + 3) is defined iff 2x + 3 ≠ 0
→ x ≠ - 3/2
→ x belongs to ] - ∞, -3/2[ U ]-3/2, +∞[
d- (x + 2)/x(x - 2) is defined iff x(x – 2) ≠ 0
→ x ≠ 0 and x ≠ 2
→ x belongs to ] - ∞, 0[ U ]0, 2[ U ]2, +∞[
e- (x2+ x + 1)/(x2 + 4x) is defined iff x2 + 4x ≠ 0
→ x(x + 4) ≠ 0
→ x ≠ 0 and x ≠ -4
→ x belongs to ] - ∞, -4[ U ]-4, 0[ U ]0, +∞[
f- (2 x + 1)/(x2 - 4) is defined iff x2 - 4 ≠ 0
→ (x – 2)(x + 2) ≠ 0
→ x ≠ 2 and x ≠ -2
→ x belongs to ] - ∞, -2[ U ]-2, 2[ U ]2, +∞[
14a- x(x + 1) / x(x – 2)
x(x – 2) ≠ 0 → x ≠ 0 and x ≠ 2
→ Df: ]-∞,0[ U ]0, 2[ U ]2, +∞[
Simplify: x(x + 1) / x(x – 2) = (x + 1)/(x – 2)
b- x(x + 2) / 3(x + 2)
x + 2 ≠ 0 → x ≠ -2
→ Df: ]-∞, -2[ U ]-2, +∞[
Simplify: x(x + 2) / 3(x + 2) = x/3
c- x + 3 / x(x + 3)
x(x + 3) ≠ 0 → x ≠ 0 and x ≠ -3
→ Df: ]-∞,-3[ U ]-3, 0[ U ]0, +∞[
Simplify: (x + 3) / x(x + 3) = 1/x
d- (2x – 1)(x - 5) / (4x – 2)(x + 6)
(4x – 2)(x + 6) ≠ 0 → x ≠ 1/2 and x ≠ -6
→ Df: ]-∞,-6[ U ]-6, 1/2[ U ]1/2, +∞[
Simplify: (2x – 1)(x - 5) / (4x – 2)(x + 6) = (2x – 1)(x - 5) / 2(2x – 1)(x + 6) = (x – 5) / 2(x + 6)
e- (x + 1)(x - 5) / (5 – x)(x + 3)
(5 – x)(x + 3) ≠ 0 → x ≠ 5 and x ≠ -3
→ Df: ]-∞,-3[ U ]-3, 5[ U ]5, +∞[
Simplify: (x + 1)(x - 5) / (5 – x)(x + 3) = (x + 1)(x - 5) / -(x – 5)(x + 3) = (x + 1) / -(x + 3)
f- (x – 1) / (x2 – 1) = (x – 1) / (x – 1)(x + 1)
(x – 1)(x + 1)≠ 0 → x ≠ -1 and x ≠ 1
→ Df: ]-∞,-1[ U ]-1, 1[ U ]1, +∞[
Simplify: (x – 1) / (x – 1)(x + 1) = 1 / (x + 1)
15a- p(x) = x2 – 4 + x(x + 2)
= (x – 2)(x + 2) + x(x + 2)
= (x + 2)[(x – 2) + x]
= (x + 2)(2x – 2)
q(x) =(x – 1)2 + 4x - 4
= (x – 1)2 + 4(x - 1)
= (x - 1)[(x - 1) + 4]
= (x - 1)(x + 3)
p(x) / q(x) = (x + 2)(2x – 2) / (x - 1)(x + 3)
(x - 1)(x + 3) ≠ 0 → x – 1 ≠ 0 and x + 3 ≠ 0
→ x ≠ 1 and x ≠ -3
p(x) / q(x) = (x + 2)(2x – 2) / (x - 1)(x + 3)
= (x + 2)2(x – 1) / (x – 1)(x + 3)
= 2(x + 2) / (x + 3)
= (2x + 4) / (x + 3)
b- p(x) = x3 – x + 4x(x + 1)
= x(x2 – 1) + 4x(x + 1)
= x(x - 1)(x + 1) + 4x(x + 1)
= x(x + 1)[(x – 1) + 4]
= x( x + 1)(x + 3)
q(x) = 2x + 6 + (x2 – 9)
= 2(x + 3) + (x – 3)(x + 3)
= (x + 3)[2 + (x – 3)
= (x + 3)(x – 1)
p(x) / q(x) = x( x + 1)(x + 3)/ (x + 3)(x – 1)
(x - 1)(x + 3) ≠ 0 → x – 1 ≠ 0 and x + 3 ≠ 0
→ x ≠ 1 and x ≠ -3
p(x) / q(x) = x( x + 1)(x + 3)/ (x + 3)(x – 1)
= x(x + 1) / (x – 1)
= x2 + x / x - 1
c- p(x) = x2 + x + 3(x + 1)
= x(x + 1) + 3(x + 1)
= ( x + 1)(x + 3)
q(x) = x2 – 9 + 4(x + 3)2
= (x + 3)(x – 3) + 4(x + 3)2
= (x + 3)[(x – 3) + 4(x + 3)]
= (x + 3)(5x + 9)
p(x) / q(x) = ( x + 1)(x + 3)/ (x + 3)(5x + 9)
(x + 3)(5x + 9) ≠ 0 → 5x + 9 ≠ 0 and x + 3 ≠ 0
→ x ≠ -9/5 and x ≠ -3
p(x) / q(x) = ( x + 1)(x + 3)/ (x + 3)(5x + 9)
= (x + 1) / (5x + 9)
d- p(x) = (x – 1)(x2 + 2x + 1)
= (x - 1)(x + 1)2
q(x) = x3 – 2x2 - x + 2
= x2(x – 2) – (x - 2)
= (x - 2)(x2 - 1)
= (x - 2)(x – 1)(x + 1)
p(x) / q(x) = (x - 1)(x + 1)2 /(x - 2)(x – 1)(x + 1)
(x - 2)(x - 1)(x + 1) ≠ 0 → x - 2 ≠ 0 and x - 1 ≠ 0 and x + 1 ≠ 0
→ x ≠ 2 and x ≠ 1 and x ≠ -1
p(x) / q(x) = (x - 1)(x + 1)2 /(x - 2)(x – 1)(x + 1)
= (x + 1) / (x - 2)
16a- (2x + 1) / (x + 4) = 0
x + 4 ≠ 0 → x ≠ -4
→ Df : ]-∞, -4[ U ]-4, +∞[
(2x + 1) / (x + 4) = 0
2x + 1 = 0 → x = -1/2
b- x(x + 2) / (x + 3) = 0
x + 3 ≠ 0 → x ≠ -3
→ Df : ]-∞, -3 [ U ]-3, +∞[
x(x + 2) / (x + 3) = 0
x(x + 2) = 0 → x = 0 or x = -2
c- (x + 1)(x - 3) / (x - 3)(2x + 7) = 0
(x - 3)(2x + 7) ≠ 0 → x ≠ 3 and x ≠ -7/2
→ Df : ]-∞, -7/2 [ U ]-7/2, 3[ U ]3, +∞[
(x + 1)(x - 3) / (x - 3)(2x + 7) = 0
(x + 1)(x - 3) = 0 → x = -1 or x = 3 (rejected)
d- (x2 - 1) / x(x + 6) = 0
x(x + 6) ≠ 0 → x ≠ 0 and x ≠ -6
→ Df : ]-∞, -6 [ U ]-6, 0[ U ]0, +∞[
(x2 - 1) / x(x + 6) = 0
(x2 - 1) = 0 → (x – 1)(x + 1) = 0 → x = -1 or x = 1
e- (x2 - 4) / (x – 2)(x + 4) = 0
(x – 2)(x + 4) ≠ 0 → x ≠ 2 and x ≠ -4
→ Df : ]-∞, -4 [ U ]-4, 2[ U ]2, +∞[
(x2 - 4) / (x – 2)(x + 4) = 0
(x2 - 4) = 0 → (x – 2)(x + 2) = 0 → x = -2 or x = 2 (rejected)
f- (x + 1)(x + 5) / x(x2 - 1) = 0
x(x2 - 1) ≠ 0 → x ≠ 0 and x ≠ -1 and x ≠ 1
→ Df : ]-∞, -1 [ U ]-1, 0[ U ]0, 1[ U ]1, +∞[
(x + 1)(x + 5) / x(x2 - 1) = 0
(x + 1)(x + 5) = 0 → x = -5 or x = -1 (rejected)
g- 2x / (x + 2) = 1
x + 4 ≠ 0 → x ≠ -2
→ Df : ]-∞, -2[ U ]-2, +∞[
2x / (x + 2) = 1
2x = x + 2 → x = 2
h- 2x / (x - 3) = 2/3
x-3≠0 → x≠3
→ Df : ]-∞, 3[ U ]3, +∞[
2x / (x - 3) = 2/3
6x = 2(x – 3) → 4x = -6 → x = -3/2
i- x / (x + 2) = 4/5
x + 2 ≠ 0 → x ≠ -2
→ Df : ]-∞, -2[ U ]-2, +∞[
x / (x + 2) = 4/5
4(x + 2) = 5x → 4x + 8= 5x → x = 8
j- x / (x + 1) = (2x – 1)/(2x + 3)
x + 1 ≠ 0 → x ≠ -1 and 2x + 3 ≠ 0 → x ≠ -3/2
→ Df : ]-∞, -3/2[ U ]-3/2, -1[ U ]-1, +∞[
x / (x + 1) = (x + 1)/(2x - 1)
x(2x + 3) = (x + 1)(2x – 1) → 3x - x= -1 → x = -1/2
Chapter 11
Problems page 131
1- p(x) = (x – 5)2 – 2(x – 5)(x + 3)
a- p(x) = x2 – 10x + 25 – 2x2 + 4x + 30
= -x2 – 6x + 55
b- p(x) = (x – 5)2 – 2(x – 5)(x + 3)
= (x – 5)[(x – 5) – 2(x + 3)]
= (x – 5)(-x – 11)
c- p(√5) = -(√5)2 - 6√5 + 55
= -5 - 6√5 + 55
= 50 - 6√5
d- (x – 5)(x + 1) = 0
x – 5 = 0 or x + 1 = 0
x = 5 or x = -1
2- E(x) = (3x – 2)2 – 16
a- E(√3) = (3√3 – 2)2 – 16
= (3√3)2 – 2(3√3)(2) + 4 – 16
= 27 - 12√3 + 4 – 16
= 15 - 12√3
b- E(x) = (3x – 2)2 – 16
= [(3x – 2) – 4][(3x – 2) + 4]
= (3x – 6)(3x + 2)
c- (3x – 6)(3x + 2) = 0
3x – 6 = 0 or 3x + 2 = 0
x = 2 or x = -2/3
3- Q(x) = (x – 2)(3x + 1) – (x – 2)(x + 6)
a- Q(x) = 3x2 + x – 6x – 2 – x2 – 6x + 2x + 12
= 2x2 -9x + 10
b- Q(x) = (x – 2)[(3x + 1) – (x + 6)]
= (x – 2)(2x – 5)
c- Q(x) = 10
2x2 – 9x + 10 = 10
2x2 – 9x = 0
x(2x – 9) = 0
x = 0 or x = 9/2
4a- (2x + 1)2 – (3x – 4)(1 + 2x)
= 4x2 + 4x + 4 – (3x + 6x2 – 4 – 8x)
= 4x2 + 4x + 4 + 5x – 6x2 + 4
= -2x2 + 9x + 5
b- (2x + 1)2- (3x – 4)(1 + 2x)
= (2x + 1)[(2x + 1) – (3x – 4)]
= (2x + 1)(-x + 5)
c- A(1.5) = -2(1.5)2 + 9(1.5) + 5
= -2(3.75) + 13.5 + 5
= -7.5 + 13.5 + 5
= 11
A(√2) = -2(√2)2 + 9√2 + 5
= -4 + 9√2 + 5
= 1 + 9√2
d- A(x) = 0
(2x + 1)(-x +5) = 0
2x + 1 = 0 or -x + 5 = 0
x = -1/2 or x = 5
5a- E(x) = (4x – 3)2 + 6x(4 – x) – (x2 + 9)
= 16x2 – 2(4x)(3) + 24x – 6x2 – x2 – 9
= 9x2
= (3x)2
b- E(x) = 144
9x2 = 144
9x2 – 144 = 0
(3x – 12)(3x +12) = 0
3x – 12 = 0 or 3x + 12 = 0
x = 4 or x = -4
c- E(√3/3) = 9(√3/3)2 = 9 x 3/9 = 3
6a- E(x) = (3/2x – 1/3)2 – 4/9
= 9/4x2 – 2(3/2x)(1/3) + (1/3)2 – 4/9
= 9/4x2 – x – 1/3
b- E(x) = [(3/2x – 1/3) – (2/3)] [(3/2x – 1/3) + (2/3)]
= (3/2x – 1)(3/2x + 1/3)
c- E(x) = (3/2x – 1)(3/2x + 1/3) = 0
3/2x – 1 = 0 or 3/2x + 1/3 = 0
x = 3/2 or x = -2/9
7a- A = √2 + 3 – (6√2 – 4)
= √2 + 3 - 6√2 + 4
= -5√2 + 7
B = (√2 + 3)(6√2 – 4)
= 12 - 4√2 + 18√2 – 12
= 14√2
b- (√2 + 3) / (6√2 – 4) → rationalize
(√2 + 3) / (6√2 – 4) x (√2 + 3) / (6√2 – 4)
= (12 + 4√2 + 18√2 + 12) / (72 – 16)
= (12 + 11√2) / 28
c- (√2 + 3) – (6√2 – 4) = -5√2 + 7 < 0
→ √2 + 3 < 6√2 – 4
8a- x2 – (x + 5)(x – 5) = x2 - (x2 – 25)
= 25
b- (8769645610)2 - 8769645615 x 8769645605
= (8769645610)2 - (8769645610 + 5) (8769645610 – 5)
Let x = 8769645610
→ x2 - (x + 5)(x – 5) = 25
9a- x2 = (√2(1 + √6))2
= (√2)2(1 + √6)2
= 2(1 + 2√6 + 6)
= 14 + 4√6
y2 = (2 - √6)2
= 4 - 4√6 + 6
= 10 - 4√6
x2 + y2 = 14 + 4√6 + 10 - 4√6 = 24
b- h2 = x2 + y2 = 24
→ h = √24 = 2√6
10a- A(2/5) = 5(2/5)2 – 12(2/5) + 4
= 5(4/25) – 24/5 + 4 = 0
b- 5(2)2 – 12(2) + 4 = 20 – 24 + 4 = 0
→ 2 is a solution
c- (5x – 2)(x – 2) = 0
5x – 2 = 0 or x – 2 = 0
x = 2/5 or x = 2
d- (5x – 2)(x – 2) = 5x2 – 10x – 2x + 4
= 5x2 – 12x + 4
= A(x)
11a- (5x + 3)(x + 1) = 5x2 + 5x + 3x + 3
= 5x2 + 8x + 3
b- (3x + 2)2 – (2x + 1)2 = [(3x + 2) – (2x + 1)] [(3x + 2) + (2x + 1)]
= (x + 1)(5x + 3)
c- BC2 = AB2 + AC2
AC2 = BC2 – AB2 = (3x + 2)2 – (2x + 1)2
= (x + 1)(5x + 3)
= 5x2 + 8x + 3
d- x = 3
AB = 2x + 1 = 2(3) + 1 = 7
AC2 = 5x2 + 8x + 3 = 5(3)2 + 8(3) + 3 = 45 + 24 + 3 = 72
→ AC = √72 = 6√2
BC = 3x + 2 = 3(3) + 2 = 11
12a- E(x) = (x2 – 4) / (x + 2)
x + 2 ≠ 0 → x ≠ -2
→ E(x) = (x – 2)(x + 2) / (x + 2) = x – 2
b- In the case when x = -2, then E(x) is not defined
13a- A = l x w
= (x2 + 10x + 25)/(x2 – 49) x (x + 7)/(x2 – 25)
= (x + 5)2(x + 7) / (x – 7)(x + 7)(x – 5)(x + 5)
= (x + 5) / (x-7((x – 5)
b- A = (x3 – 4x)/(x2 + 4x + 4) x (2x + 4)/x2
= 2x(x2 – 4)(x + 2) / (x + 2)2(x2)
= 2x(x – 2)(x + 2)2 / (x +2)2(x2)
= 2(x – 2) / x
14a- A = (x2 – 1) /2 and w = x – 1
A = l x w → (x2 – 1) / 2 = l x (x – 1)
→l = (x2 – 1)/2 / (x – 1) = (x + 1) / 2
b- Square → l = w
(x + 1) / 2 = x – 1 →x + 1 = 2x – 2
→ x=3
15L = w + 20
Area A = l x w = (w + 20)(w) = w2 + 20w
w’ = w + 16
l’ = l – 10 = w + 20 – 10 = w + 10
A’ = A = l’ x w’
= (w + 10)(w + 16) = w2 + 20w
= w2 + 16w + 10w + 160 = w2 + 20w
→ 26w – 20w + 160 = 0
6w = -160 → w = -80/3 impossible (negative width)
1614x3 – 16x2 – 17x + 30
x=3
→ 14(3)3 – 16(3)2 – 17(3) + 30 = 213
17a- Area(big rectangle) = l x w = A1
= 3x(4x + 2) = 12x2 + 6x
Area(small rectangle) = l x w = A2
= 2x(x) = 2x2
Area(green region) = A1 – A2
= 12x2 + 6x – 2x2
= 10x2 + 6x
b- Area(big rectangle) = A1 = l x w
= (2x + 5)(2x)
= 4x2 + 10x
Area(small rectangle) = A2 = l x w
= 5 x 3 = 15
Area(green region) = A1 –A2
= 4x2 + 10x – 15
c- Area(big square) = A1 = s x s
= (4x)(4x) = 16x2
Area(small square) = A2 = s x s
= (x – 1)(x) = x2 – x
Area(green region) = A1 – A2
= 16x2 – x2 + x
= 15x2 + x
18Area(LAND) = l x w = (3 + x)(2x + 5 +3)
= (3 + x)(2x + 8)
= 2x2 + 14x +24
Area(yellow) = l x w = (2x + 5)(x)
= 2x2 + 5x
Area(blue) = (2x2 + 14x + 24) – (2x2 + 5x)
= 9x + 24
9x + 24 = 213 → x = 21
w = 3 + x = 3 + 21 = 24
l = 2x + 8 = 2(21) + 8 = 50
19Area(square) = s x s = (4x)(4x) = 16x2
Area(circle) = πx2 = 3.14x2
Area(4 circles) = 4(3.14x2) = 12.56x2
Area(red region) = 16x2 – 12.56x2 = 3.44x2