Chapter 6More about Polynomials 1
Chapter 6More about Polynomials
Warm-up Exercise1.Let f(x) 2x2x 3, find the values of the following.
(a)f(2)(b)
2.Let g(x) 4x2 5x 6, find the values of the following.
(a)g(3)(b)
3.Let h(x) 2x3 6x2 8x 5, find the values of the following.
(a)h(5)(b)
4.Factorize the following expressions.
(a)x2 5x 4(b) 2x2 2x 24
5.Factorize the following expressions.
(a)3x2 7x 6(b)6x2 7x 20
6.(a)If (3x 2)2Ax2BxC, find the values of A, B and C.
(b)If (2x 1)(3x 5)Ax2BxC, find the values of A, B and C.
7.(a)If 2x2x 2 A(x 1)2Bx, find the values of A and B.
(b)If A(2x 1)2B(4x 1) 16x2, find the values of A and B.
8.Solve the following simultaneous equations.
(a)(b)
9.Solve the following simultaneous equations.
(a)(b)
Build-up Exercise[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set, Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]
Exercise 6A
Elementary Set
Level 1
Simplify the following. (110)
1.(2x2 6x 5) (3x2 2x 1)2.(3x2 2x 9) (10x2 11x 22)
3.(12x2 7x 2) (9x2 10x 21)4.(x3 4x2 6x 3) (2x3 5x2 8x 6)
5.(8x3 2x2 14) (11x3 4x)6.(5x2 7x 10) (2x2 6x 5)
7.(11x2 24x 5) (6x2 2x 1)8.(30x2 45x 9) (40x2 60x 21)
9.(2x3 15x2 12x 19) (x3 10x2 6x 5)10.(3x3 16x 5) (8x3 12x2 9)
Expand the following. (1119)
11.x(3 x)12.2x(x 2)
13.(x 1)(x 5)14.(2x 3)(4x 5)
15.x(x2 1) 2(x2 1)16.x(2x2 3) 3(2x2 3)
17.(3x2 1)(2x 3)18.(4x2 2)(x 1)
19.(6x2 3x)(2x 7)
Level 2
Expand the following. (2023)
20.(2x2 3x 1)(2x 1)21.(3x2 6x 2)(3x 4)
22.(3 4x)(2 7x2 4x)23.(6 3x 2x2)(3 5x)
Simplify the following. (2425)
24.(x2 1)(x 3) (1)25.(9x2 5x 2)(2x 1) 6x 9
Intermediate Set
Level 1
Simplify the following. (2633)
26.(2x2 5x 4) (9x2 11x 22)27.(8x2 6x 7) (2x2 9x 5)
28.(x3 4x2 6x 3) (2x3 6x2 4x 6)29.(5x3 2x2 7x 10) (6x3 2x 5)
30.(11x2 24x 5) (6x2 2x 3)31.(9x3 12x2 6x 7) (5x3 6x2 7x 8)
32.(8x3 2x2 7x) (4x3 6x 2)33.(7x3 8x2 6x 2) (5x3 12x2 9x 11)
Expand the following. (3439)
34.(4x2 1)(x 2)35. (2x2x)(3x 4)
36.(2x 9)(2x2 3)37.(5x2 6)(2x 3)
38.(x2 2x)(x 1)39.(3x2 2)(3x 4)
Level 2
Expand the following. (4042)
40.(5x2 7x 10)(5 2x)41.(6x2 7x 4)(3 5x)
42.(3x2 5 6x)(5 4x)
Simplify the following. (4350)
43.(3x2 2x 4)(x 3) 644.(2x2 3 5x)(2 x) 2x
45.(8x2 5x 2)(2x 1) 5x 946.(5x2 3x 1)(3x 2) 3(x2 6)
47.(3 4x)(3 6x2 3x) 2(9 5xx2)
48.(7x2 6x 1)(3x 5) (x 6)(2x 3)
49.(8x2 2x 9)(6x 7) (2x2 5x)(2x 6)
50.(5x2 30x 11)(7x 5) (5x2 2x 5)(x 4)
Advanced Set
Level 1
Simplify the following. (5156)
51.(8x2 6x 7) (2x2 9x 5)
52.(5x3 2x2 8x 11) (6x3 2x 5)
53.(3x3 2x2 6x 1) (7x3 8x2 11x 9)
54.(21x2 30x 4) (8x2 6x 11)
55.(9x3 12x2 6x 9) (5x3 6x2 8x 8)
56.(7x3 8x2 6x 2) (5x3 12x2 9x 11)
Expand the following. (5760)
57.(3x2x)(4x 3)58.(6x2 5)(2x 3)
59.(5x2 10)(2x 5)60.(3x2 6x)(3x 4)
Level 2
Expand the following. (6165)
61.(6 x 2x2)(4 7x)62.(x2 2x 5)(8 3x)
63.(3x2 4x 3)(2 3x)64.(8x2 7x 5)(5 4x)
65.(5x2 7x 6)(4 2x)
Simplify the following. (6675)
66.(3x2 2x 4)(x 3) 767.(2x2 3 6x)(3 x) x
68.(3x 4 x2)(2x 1) 4x69.(8x2 7x 3)(2x 1) 5x 9
70.(5x2 3x 1)(3x 2) 4(x2 2)71.(3 4x)(5 6x2 3x) 2(9 5xx2)
72.(7x2 6x 2)(3x 5) (x 6)(2x 3)
73.(9x2 2x 8)(6x 7) (2x2 5x)(2x 6)
74.(x2 2x 3)(2x 9) (2x 1)3
75.(5x2 30x 11)(7x 5) (5x2 2x 6)(2x 1)
Exercise 6B
Elementary Set
Level 1
Find the quotient and the remainder of each of the following divisions. (120)
1.(3x 4) (x 1)2.(5x 4) (x 1)
3.(4x 5) (2x 1)4.(4x 1) (2x 1)
5.(x2x 2) (x 1)6.(x2 8x 5) (x 3)
7.(x2 8x 11) (x 1)8.(x2 16x 21) (x 5)
9.(x2 8x 12) (x 8)10.(x2 22x 15) (x 9)
11.(4x2 20x 5) (2x 1)12.(6x2 8x 1) (3x 1)
13.(27x2 12x 10) (3x 2)14.(27x2 15x 10) (3x 2)
15.(x3 3x2 8x 6) (x 1)16.(x3 6x2 5x 11) (x 4)
17.(x3 2x2 7x 13) (x 2)18.(x3 9x2 10x 6) (x 2)
19.(6x3 3x2 5x 1) (2x 1)20.(8x3 12x2 2x 9) (2x 1)
21.When x2pxq is divided by x 2, the quotient is x 3 and the remainder is 4. Find the values of p and q.
22.When x2axb is divided by x 1, the quotient is x 2 and the remainder is 3. Find the values of a and b.
23.When x2mx 5 is divided by x 1, the quotient is x 3 and the remainder is n. Find the values of m and n.
Level 2
Find the quotient and the remainder of each of the following divisions. (2427)
24.(x3 6x2 15x 7) (x2x 2)25.(x3 12x2 8x 9) (x2x 1)
26.(6x3 11x2 4x 7) (2x2x 3)27.(6x3 8x2 5) (2x2 1)
28.Suppose f(x) is a polynomial, the quotient is 2x2 5x1 and the remainder is 4 when f(x) is divided by 3x 2.
(a)Find the polynomial f(x).
(b)If g(x) 2f(x) (x2 2x), find g(x).
(c)Find the remainder when g(x) is divided by x 3.
Intermediate Set
Level 1
Find the quotient and the remainder of each of the following divisions. (2941)
29.(x2 7x 5) (x 3)30.(x2 6x 5) (x 5)
31.(x2 9x 20) (x 5)32.(2x2 12x 5) (x 2)
33.(6x2 11x 1) (3x 1)34.(16x2 24x 5) (4x 3)
35.(x3 6x2 5x 11) (x 4)36.(x3 8x2 20x 4) (x 4)
37.(x3 2x2 8x 12) (x 2)38.(x3 11x 5x2 9) (x 3)
39.(6x3 5x2 33x 10) (2x 3)40.(9x3 12x2 3x 9) (3x 2)
41.(22x2 8x 9 15x3) (5x 4)
42.When x2px 7 is divided by x 2, the quotient is x 5 and the remainder is q. Find the values of p and q.
43.When 3x2pxq is divided by x 3, the quotient is rx 5 and the remainder is 6. Find the values of p, q and r.
44.When x3px2qxr is divided by x2 2x 3, the quotient is x 4 and the remainder is 2x 1. Find the values of p, q and r.
Level 2
Find the quotient and the remainder of each of the following divisions. (4554)
45.(x3 6x2 15x 7) (x2 2x 1)46.(x3 13x2 8x 7) (x2x 2)
47.(2x3 8x2 9x 12) (x2 5x 1)48.(6x3 13x2 10x 4) (x2 3 x)
49.(6x3 8x2 6) (2x2 1)50.(9x3 4 7x) (x 3x2)
51.(9 4x3x 6x2) (2x2 5)52.(x4 5x3 3x2 15x 7) (x2 2x 3)
53.(x4 7x3 52x2 28x 16) (x2 5x 4)
54. (2x4 9x2 4) (x2 4x 2)
55.Suppose f(x) is a polynomial, the quotient is 4x2 3 and the remainder is 2 when f(x) is divided by 2x 3.
(a)Find the polynomial f(x).
(b)If h(x) 4f(x) 9, find h(x).
(c)Find the quotient and the remainder when h(x) is divided by 8x 4.
56.When f(x) px3qx2 8 is divided by 3x 2, the quotient is 9x2 9x 6 and the remainder is 20.
(a)Find the values of p and q.
(b)Hence solve the equation f(x) 8.
Advanced Set
Level 1
Find the quotient and the remainder of each of the following divisions. (5766)
57.(x2 6x 4) (x 5)58.(x2 12x 5) (x 2)
59.(10xx2 39) (x 3)60.(8x2 26x 7) (4x 3)
61.(12x2 7x 21) (4x 5)62.(x3 7x2 10) (x 1)
63.(x3 6x2 12x 9) (x 3)64.(6x3 7x2 34x 9) (3 2x)
65.(22x2 15x3 6 8x) (5x 4)66.(24x3 18x 7) (3x 3)
67.When 10x2pxq is divided by x 4, the quotient is rx 2 and the remainder is 7. Find the values of p, q and r.
68.When px3qx2rx 4 is divided by 2x2x 1, the quotient is 3x 2 and the remainder is 4x 6. Find the values of p, q and r.
69.When px3qx2 14xr is divided by 4x2 5, the quotient is 3x 8 and the remainder is x 3. Find the values of p, q and r.
Level 2
Find the quotient and the remainder of each of the following divisions. (7080)
70.(x3 6x2 15x 8) (x2 2x 1)71.(x3 12x2 8x 7) (x2x 1)
72.(2x3 8x2 11x 2) (x2 5 4x)73.(x2 17x 4x3 4) (x2x 7)
74.(2 6x2 28x 5x3) (x2 4 2x)75.(2x3 2x2 5x 1) (2x2 1)
76.(12x3 10x2 3x 1) (3x2 4x)77.(8 4x3 3x 6x2) (2x2 5)
78.(x4 5x3 3x2 21x 7) (2x 3 x2)
79.(7x3 8x2 4x4 – 30x 53) (4 x2x)
80.(89x 47 3x4 40x2) (x2 4x 2)
81.Suppose f(x) is a polynomial, the quotient is 2x2 5 and the remainder is 3 when f(x) is divided by 3x 2.
(a)Find the polynomial f(x).
(b)If g(x) 6x2 2x 6f(x), find g(x).
(c)Find the quotient and the remainder when g(x) is divided by 2x 3.
82.When f(x) 4x2 12xp is divided by 2x 1, the quotient is qx 7 and the remainder is 3.
(a)Express f(x) in terms of x and q.
(b)Find the values of p and q.
(c)Hence solve the equation f(x) 6 0.
83.When g(x) px3qxr is divided by x 1, the quotient is 2x2 2x 7 and the remainder is 4.
(a)Find the values of p, q and r.
(b)Hence solve the equation g(x) 4.
84.When a polynomial f(x) is divided by x 1, the quotient is px 31 and the remainder is 33.
(a)Express f(x) in terms of x and p.
(b)When f(x) is divided by 2x 3, the quotient is 3 5x and the remainder is 7. Find the value of p.
(c)Hence solve the equation f(x) x 0, leave your answers in surd form if necessary.
Exercise 6C
Elementary Set
Level 1
In each of the following questions, use the remainder theorem to find the remainder when f(x) is divided by g(x). (18)
1.f(x) 2x2 5x 3
(a)g(x) x 1(b)g(x) x 1(c)g(x) x 2
2.f(x) 3x2 2x 5
(a)g(x) x 1(b)g(x) x 2(c)g(x) x 3
3.f(x) 4x2 6x 1
(a)g(x) x 3(b)g(x) x 4(c)g(x) x 5
4.f(x) 8x2 2x 9
(a)g(x) 2x 1(b)g(x) 2x 1(c)g(x) 2x 3
5.f(x) 18x2 6x 4
(a)g(x) 2x 1(b)g(x) 3x 1(c)g(x) 3x 2
6.f(x) 8x3 6x2 5x 1
(a)g(x) x 2(b)g(x) x 5(c)g(x) 2x 3
7.f(x) x3 6x2 9x 2
(a)g(x) 2x 1(b)g(x) 2x 3(c)g(x) 2x 5
8.f(x) 14x3 7x2 5x 3
(a)g(x) 2x 1(b)g(x) 2x 3(c)g(x) 2x 5
9.When f(x) x3 2x2k 7 is divided by x 1, the remainder is 4. Find the value of k.
10.When f(x) 27x3kx2 3x 2 is divided by 3x 1, the remainder is 1. Find the value of k.
11.When f(x) x2 (k 1)x 2 is divided by xk, the remainder is 3. Find the value of k.
12.When f(x) x2ax 4 is divided by xa, the remainder is 4a 2. Find the values of a.
In each of the following questions, determine whether f(x) is divisible by g(x). (1316)
13.f(x) x3 5x 4
(a)g(x) x 1(b)g(x) x 1(c)g(x) x 3
14.f(x) x3 2x2x 3
(a)g(x) x 2(b)g(x) 2x 1(c)g(x) 2x 1
15.f(x) 2x3x2 9
(a)g(x) x 3(b)g(x) 2x 3(c)g(x) 2x 5
16.f(x) 40 82x 9x3 27x2
(a)g(x) x 5(b)g(x) 3x 2(c)g(x) 3x 4
17.If f(x) 2x3kx2 3x 2 is divisible by x 2, find the value of k.
18.If 2x 3 is a factor of f(x) 6x3 5x2kx 18, find the value of k.
Level 2
19.Let f(x) x2axb. When f(x) is divided by x 1 and x 2, the remainders are 0 and 1 respectively. Find the values of a and b.
20.Let f(x) x3ax2bx 2. When f(x) is divided by x 1 and x 2, the remainders are 4 and 32 respectively.
(a)Find the values of a and b.
(b)Find the remainder when f(x) is divided by x2x 2.
21.It is given that x 3 is factor of f(x) x3kx2 5x 12.
(a)Find the value of k.
(b)Hence factorize f(x).
22.It is given that 2x 1 and x 2 are factors of f(x) 2x3px2qx 6.
(a)Find the values of p and q.
(b)Hence factorize f(x).
23.Let f(x) 2x3 3x2 2x 3.
(a)Prove that x 1 is a factor of f(x).
(b)Factorize f(x).
(c)Solve the equation f(x) 0.
24.It is given that f(x) ax3 2bx2x 6 and g(x) bx3 (8 a)x2 13x 6 are both divisible by x 2.
(a)Find the values of a and b.
(b)Solve the equation 2f(x) g(x).
25.It is given that x2x 6 is a factor of f(x) ax3ax2bx 12.
(a)Find the values of a and b.
(b)Solve the equation f(x) 0.
Intermediate Set
Level 1
In each of the following questions, use the remainder theorem to find the remainder when f(x) is divided by g(x). (2629)
26.f(x) 2x2 5x 4
(a)g(x) x 1(b)g(x) x 1(c)g(x) x 2
27.f(x) 8x2 4x 7
(a)g(x) 2x 1(b)g(x) 2x 1(c)g(x) 2x 3
28.f(x) 2x3 6x2 5x 1
(a)g(x) x 1(b)g(x) x 3(c)g(x) 2x 3
29.f(x) 14x3 7x2 5x 3
(a)g(x) x 1(b)g(x) 2x 1(c)g(x) 2x 3
30.When f(x) x3 3x2k 7 is divided by x 1, the remainder is 3. Find the value of k.
31.When f(x) 6x3kx2 3x 2 is divided by 3x 1, the remainder is 2. Find the value of k.
32.When f(x) x3 (k2 1)x 2 is divided by xk, the remainder is 1. Find the value of k.
33.When f(x) x3 (k 2)x2 3x 2 is divided by xk, the remainder is k2. Find the values of k.
In each of the following questions, determine whether f(x) is divisible by g(x). (3435)
34.f(x) 2x3 3x2 4x 1
(a)g(x) x 1(b)g(x) x 2(c)g(x) x 3
35.f(x) 6x3 4x2 12x 3
(a)g(x) 2x 1(b)g(x) 2x 3(c)g(x) 3x 1
36.If f(x) 2x3 3x2kx 2 is divisible by x 2, find the value of k.
37.If f(x) 3x3kx2 4x 3k is divisible by x 1, find the value of k.
38.If 1 2x is a factor of f(x) 4x3 (k 1)x2 3x 4, find the value of k.
Level 2
39.Let f(x) 2x2pxq. When f(x) is divided by x 2 and x 3, the remainders are 5 and 30 respectively. Find the values of p and q.
40.When f(x) x3px2qx 24 is divided by x 2 and x 3, the remainders are 4 and 42 respectively. Find the values of p and q.
41.When f(x) x3 4x2pxq is divided by x 1 and x 3, the remainders are 4 and 76 respectively.
(a)Find the values of p and q.
(b)Find the remainder when f(x) is divided by x2 2x 3.
42.It is given that x 2 is a factor of f(x) px3 3x2 3x 2.
(a)Find the value of p.
(b)Hence factorize f(x).
43.It is given that x 1 and 2x 3 are factors of f(x) ax3 5x2bx 9.
(a)Find the values of a and b.
(b)Hence factorize f(x).
44.Let f(x) 2x3px2 25xq. It is known that 2x 1 and x 3 are factors of f(x).
(a)Find the values of p and q.
(b)Hence factorize f(x).
45.Let f(x) 6x3 5x2 3x 2.
(a)Prove that 3x 2 is a factor of f(x).
(b)Solve the equation f(x) 0.
46.Let f(x) 2x3 5x2x 6.
(a)Prove that 3 2x is a factor of f(x).
(b)Solve the equation f(x) 2x 3, leave your answers in surd form if necessary.
47.It is given that f(x) 6x3x2axb and g(x) 2bx3 8x2 15x (a 2b) are both divisible by 2x 1.
(a)Find the values of a and b.
(b)Solve the equation f(x) g(x) 0, leave your answers in surd form if necessary.
48.It is given that (x 3)(2x 1) is a factor of f(x) 2x3 3ax2 (a 2b)x 6.
(a)Find the values of a and b.
(b)Solve the equation f(x) 2(x 3).
49.Without doing an actual division, find the remainder when x3 3x2 4x 5 is divided by x(x 1).
50.What number should be added to the polynomial 2x3 3x2 5x 2 so that the resulting polynomial is divisible by x 1?
Advanced Set
Level 1
In each of the following questions, use the remainder theorem to find the remainder when f(x) is divided by g(x). (5153)
51.f(x) 3x3 4x 5
(a)g(x) x 1(b)g(x) x 1(c)g(x) x 2
52.f(x) 8x3 7x2x 4
(a)g(x) 2x 1(b)g(x) 2x 1(c)g(x) 2x 3
53.f(x) 8x2 15x3 3x 2
(a)g(x) x 2(b)g(x) 3x 1(c)g(x) 5x 1
In each of the following questions, determine whether f(x) is divisible by g(x). (5456)
54.f(x) 2x3x2 11x 10
(a)g(x) x 1(b)g(x) x 2(c)g(x) 2x 3
55.f(x) 6x3 3x2x 12
(a)g(x) 2x 1(b)g(x) 2x 1(c)g(x) 2x 3
56.f(x) 15x3 49x2 4x 3
(a)g(x) 2x 1(b)g(x) 3x 1(c)g(x) 5x 2
57.When f(x) x3kx2 2kx 8 is divided by xk, the remainder is 16. Find the values of k.
58.It is given that f(x) 2x3 (2k 1)x2 4x 3 is divisible by xk. Find the values of k.
Level 2
59.It is given that x 3 is a common factor of
f(x) 4bx2ax 45 and g(x) x3ax2bx 3.
Find the values of a and b.
60.When f(x) ax3bx2 7x 15 is divided by x 2 and x 3, the remainders are 5 and 90 respectively. Find the values of a and b.
61.When f(x) ax3 8x2bx 5 is divided by x 3 and x 4, the remainders are 5 and 29 respectively.
(a)Find the values of a and b.
(b)Find the remainder when f(x) is divided by x2x 6.
62.When f(x) 2ax3x2bxb is divided by 2x 1 and 2x 3, the remainders are 5 and 50 respectively.
(a)Find the values of a and b.
(b)Find the remainder when f(x) is divided by 2x2x 3.
63.It is given that x 2 and 2x 3 are factors of f(x) 2x3px2qx 6.
(a)Find the values of p and q.
(b)Hence factorize f(x).
64.It is given that f(x) 6x3px2 26xq is divisible by 3x 2 and x 3.
(a)Find the values of p and q.
(b)Hence factorize f(x).
65.It is given that f(x) 2px3 (q 1)x2 17xp is divisible by 2x 3 and 2x 1.
(a)Find the values of p and q.
(b)Hence factorize f(x).
66.It is known that x 2 is a factor of f(x) x3kx2x 2.
(a)Find the value of k.
(b)Solve the equation f(x) 0.
67.Let f(x) 6x3 7x2 6x 1.
(a)Prove that 3x 1 is a factor of f(x).
(b)Solve the equation f(x) 3x 1.
68.When f(x) 2x3px2 7xq is divided by x and x 1, the remainders are both 6.
(a)Find the values of p and q.
(b) Solve the equation f(x) 6.
69.It is known that f(x) ax3 10x2bx (3b 2a) and g(x) 9x3 (4b 1)x2 (2ab)x 3 are both divisible by 3x 1.
(a)Find the values of a and b.
(b)Solve the equation f(x) 2g(x) 0, leave your answers in surd form if necessary.
70.It is given that 3x2 11x 4 is a factor of f(x) ax3 (b 2)x2 2bx 24.
(a)Find the values of a and b.
(b)Solve the equation f(x) 3(3x 1).
71.Without doing an actual division, find the remainder when 2x3x2 9is divided by (x 1)(x 2).
72.Without doing an actual division, find the remainder when 6x3 13x2 36x 57is divided by 2x2 9x 9.
73.(a)Find the remainder when x2 004 is divided by x 1.
(b)Express x2 004 in terms of x 1 and Q(x), where Q(x) is the quotient of x2 004 (x 1).
(c)Hence find the remainder when 42 004 3 is divided by 5.
74.What number should be added to the polynomial 10x3 14x2 57x 37so that the resulting polynomial is divisible by 5x 3?
75.Find a linear polynomial such that when it is added to the polynomial 6x3 5x2 23x 1,the resulting polynomial is divisible by 2x2 5x 2.
Exercise 6D
Elementary Set
Level 1
Factorize the following polynomials. (18)
1.x3 2x 12.x3 3x 2
3.x3 10x 114.x3x2x 1
5.x3 8x2 17x 106.x3 6x2x 6
7.x3 3x2 2x 88. x3 8x2 16x 8
Determine whether each of the following polynomials has a linear factor with integral coefficient and constant term. (914)
9.x3 3x2 3x 110.x3x2 2
11.2x3x2 3x 112.4x3 8x2 5x 1
13.17x 6 x3 5x214.3x2 6x 8 x3
Level 2
Factorize the following polynomials. (1518)
15.3x3 17x2 9x 516.4x3 8x2x 3
17.2x3 7x2 27x 1818.6x3 19x2 11x 6
19.Let f(x) 3x3 5x2 26x 8.
(a)Factorize f(x).
(b)Solve the equation f(x) 0.
20.Let g(x) 6x3 11x2 13x 15.
(a)Factorize g(x).
(b)Solve the equation g(x) 0.
Intermediate Set
Level 1
Factorize the following polynomials. (2126)
21.x3 11x 1222.x3 10x2 11
23.x3 2x2 19x 2024.x3x2 10x 8
25.3x3 7x2 5x 126.24x 9x2 20 x3
Determine whether each of the following polynomials has a linear factor with integral coefficient and constant term. (2730)
27.x3 12x2 48x 6428.4x3 3x 1
29.6x3 4x2 7x 130.14x2 24x3x 1
Level 2
Factorize the following polynomials. (3137)
31.3x3 19x2 21x 532.5x3 17x2 11x 6
33.2x3 5x2 11x 434.6x3 5x2 3x 2
35.24x3 22x2x 236.3x3 12x 32 8x2
37.8x 12 10x3 15x2
38.Let f(x) 3x3 20x2 39x 18.
(a)Factorize f(x).
(b)Solve the equation f(x) 0.
39.Let f(x) 6x3 7x2 14x 8.
(a)Factorize f(x).
(b)Solve the equation f(x) 0.
40.Solve the equation 4x3 8x2 9x 18 0.
Advanced Set
Level 1
Factorize the following polynomials. (4144)
41.x3 5x2 2x 842.2x3 7x2 5x 1
43.6x3 6x 1 11x244.19x2 30x3 1
Determine whether each of the following polynomials has a linear factor with integral coefficient and constant term. (4546)
45.8x3 4x 146.5x2 12x3 1
Level 2
Factorize the following polynomials. (4756)
47.2x3 5x2 11x 548.3x3 10x2 9x 4
49.6x3 7x2x 250.5x3 12x2 9x 2
51.2x3 11x2 18x 952.3x3 8x2 12x 32
53.4x3 4x2 19x 1054.10x3 12 41x 29x2
55.12x2 8x3 26x 1556.18x 13x2 8x3 9
57. Let f(x) 8x3 20x2 6x 9.
(a)Factorize f(x).
(b)Solve the equation f(x) 0.
58.Let g(x) 8x3 22x2 27x 15.
(a)Factorize g(x).
(b)Solve the equation g(x) 0.
59.Solve the equation 6x3x2 14x 8 0, leave your answers in surd form if necessary.
60.The figure shows the graph of yx3 6x2axb.
(a)Find the values of a and b.
(b)Factorize x3 6x2axb.
(c)Factorize f(x 1) where f(x) x3 6x2axb.
(d)Solve the equation f(x 2) x.
Chapter Test / (Time allowed: 1 hour)Section A
1.Find the remainder when 2x3 3x2 7 is divided by 2x 1.(2 marks)
2.Find the value of k if x3kx2 11x 10 is divisible by x 2. (3 marks)
3.When x3kx2x 8 is divided by x 2, the remainder is 6, find the value of k. (3 marks)
4.Let f(x) (x 2)(x 3) 3. When f(x) is divided by xk, the remainder is k. Find the values of k. (5 marks)
5.Let f(x) 2x3x2 13x 6.
(a)Find the value of f(2). (1 mark)
(b)Hence factorize f(x).(4 marks)
6.Let h(x) 2x3 15x2 34xk.
(a)If h(x) is divisible by x 2, find the value of k.(3 marks)
(b)(i)Find the value of h(4).
(ii)Hence factorize h(x). (4 marks)
Section B
7.Let g(x) px3x2qx 6. It is divisible by x 2 and has a remainder of 30 when divided by x 3.
(a)Find the values of p and q. (6 marks)
(b)Solve the equationg(x) 0. (4 marks)
8.Let f(x) 4x3px2 7x 2 and g(x) 2x3 5x2qx 8. When f(x) and g(x) are divided by x 3, the remainders are 19 and 19 respectively.
(a)Find the values of p and q. (4 marks)
(b)Solve the equation f(x) g(x) 0. (6 marks)
Multiple Choice Questions (3 marks each)
Chapter 6More about Polynomials 1
9.If A(x2Bx) 4x2 8x, then
A.A 2 and B 4.
B.A 4 and B 2.
C.A 4 and B 4.
D.A 4 and B 8.
10.When a polynomial f(x) is divided by 3x 2, the remainder is
A..
B..
C..
D..
11.Which of the following is a factor of 8x3 12x 9?
A.2x 1
B.2x 1
C.2x 3
D.2x 3
12.If ax3bxc is divisible by x 1, then
A.abc 0.
B.abc 0.
C.abc 0.
D.abc 0.
13.If x 1 and 2x 3 are factors of f(x)6x3px2qx 15, which of the following is also a factor of f(x)?
A.3x 5
B.3x 5
C.6x 5
D.6x 5
14.Let f(x) be a polynomial. If
f(0) f(2) 0,
which of the following must not be a factor of f(x)?
I.x
II.x – 2
III.x + 2
A.I only
B.II only
C.III only
D.None of the above
15.If f(x) is divisible by x 1, which of the following must be a factor of f(x 1)?
A.x
B.x 1
C.x 2
D.x 2
16.When a polynomial f(xk) is divided by xk, the remainder is
A.f(0).
B.f(k).
C.f(2k).
D.f(2k).
17.If xa is a factor of f(x), which of the following must be a factor of f(x)?
A.xa
B.xa
C.x 2a
D.x 2a
18.When polynomials f(x) and g(x) are divided by xa, the remainders are both R. Which of the following expressions must have a factor xa?
A.f(x) g(x)
B.f(x) g(x)
C.f(x) g(x)
D.f(x) g(x)
Chapter 6More about Polynomials 1
Hints / (for questions with in the textbook)Exercise 6C
27.(c)Key information
Result obtained in (b), i.e. x2 000x = (x – 1)Q(x) 2
Analysis
82 000 does not appear in any given information of the question.
We try to see if the result obtained in (b) can be used to handle the term 82 000.
Method
As the term x2 000 in the result obtained in (b) has the same structure as 82 000, we substitute x 8 into the identity and obtain 82 000 8 7Q(8) 2. To ensure a correct answer, Q(8) must be an integer.
Revision Exercise 6
33.(b)Key information
f(x) = (x – 2)Q1(x) 7, where Q1(x) is the quotient of f(x) (x 2).
f(x) = (x 1)Q2(x) – 2, where Q2(x) is the quotient of f(x) (x 1).
From (a), we obtain f(x) = (x – 2)(x 1)Q3(x) 3x 1,
where Q3(x) is the quotient of f(x) [(x 2)(x 1)].
Analysis
The required division is f(x 3) [(x 1)(x 4)] and the divisor is of degree two.
We try to see if the information f(x) = (x – 2)(x 1)Q3(x) 3x 1, which also represents a division with a divisor of degree two, can be used.
Method
As f(x) = (x – 2)(x 1)Q3(x) 3x 1 represents a division similar to the required division, it is useful to replace x with x 3 and check if this replacement gives us an identity representing the required division.