FHMM1014 MATHEMATICS 1

TUTORIAL 2 POLYNOMIAL
2.1 Remainder Theorem and Factor Theorem
1.

Find the remainder when the following polynomial is divided by the given linear function. (a)
(c)

x 4  3 x 3  5 x;

(e)
2.

x 3  2 x  4;

x 2  ax  b;

x 1
2x  1 xc (b)

x 3  3 x 2  6 x  2;

(d)

9 x 5  5 x 2  4;

(f)

x 3  2 x 2  6;

x2

3x  1 xa Factorize completely the following expressions.
(a)

x3  2x 2  x  2

(b)

x3  x 2  x  2

(c)

x4  1

(d)

x 4  x 3  3x 2  4 x  4

(e)

2x3  x 2  2x  1

3.

By using long division or synthetic division, find the quotient if x 3  4 x 2  5 x  2 divided by (x − 2).

4.

Find the remainder when the polynomial 2 x 4  5 x 3  x 2  7 x  1 is divided by (2x + 1).

5.

Find the quotient and remainder when g ( x)  3 x 3  4 x  5 is divided by f ( x)  x 2  1 .

6.

Let f ( x)  3 x 2  1 and g ( x)  5 x 4  3 x 2  6 . Find the quotient and remainder when g (x) is divided by f (x) .

7.

When the polynomial P (x) is divided by (x − 1), its remainder is 5. When
P (x) is divided by (x − 2), its remainder is 7. Given that P (x) may be written in the form (x − 1)(x − 2) · Q(x) + Ax + B, where Q (x) is a polynomial, A and
B are constants, find the remainder when P (x) is divided by (x − 1)(x − 2).

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FHMM1014 MATHEMATICS 1
8.

Show that  x  2 is a factor of f ( x)  6 x 3  13 x 2  4 . Hence, factorize f (x) completely. State the zeros of f (x) .

9.

Given that  x  2 is a factor of f (x) , where f ( x)  ax 3  10 x 2  bx  2 , a, b  R .
When f (x) is divided by (x − 3), its remainder is 16. Find the values of a and b.

10.

Given that f ( x)  x 4  ax 3  bx 2  12 x  16 where a and b are constants. If
(x − 4) is a factor of f (x) and (x − 3) is a factor f ' ( x) , show that a = −4 and determine the value of b. With these values of a and b,
(a)

find a factor of f (x) in the form (x + k) where k is a positive integer.

(b)

find all four roots of the equation f ( x)  0 .

[Note: f (x) is the derivative of f (x) .]
11.

Polynomial P(x) = 2x4 + x3 + ax2 + bx – 6 has factors (x – 2) and (2x + 1).
(i)
(ii)

12.

Find the values of a and b.
Hence, find all the four roots of the equation P(x) = 0.

The polynomial f ( x)  4 x 3  hx 2  3hx  k has a factor ( x  1) . When f (x) is divided by ( x  1) , the remainder is 10.
(i)
(ii)

13.

Find the values of h and k.
Factorize f (x) completely and solve the equation f ( x)  0 .

Polynomial f(x) = x6 – 5x4 – 10x2 + k has a factor (x – 1).
(i)
(ii)

Find the value of k.
With the value of k in part (i), find another factor in the form (x + a), where a is a constant.

14.

Given polynomial g ( x)  x 3  ( p  4) x 2  ( p  9) x  4 .
(i)

Show that the polynomial g(x) divisible by ( x  1) for all values of p.

(ii)

Find the value of p if the polynomial g(x) has remainder 12 when divided by ( x  2) .

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FHMM1014 MATHEMATICS 1

2.2 Polynomial Equations
15.

Solve the following equations.
(a)

(b)

6 x 3  19 x 2  19 x  4  0

(c)

3 x 3  2 x 2  17 x  12  0

(d)

6 x 4  25 x 3  12 x 2  25 x  6  0

(e)

16.

4 x 3  3x 2  9 x  2  0

x 4  x3  4x 2  x  1  0

(f)

3 x 4  4 x 3  14 x 2  4 x  3  0

Find all the real values of a and b that satisfy the equation







x 4  10 x 2  1  x 2  ax  1 x 2  bx  1 .
Hence determine all the real roots of the equation x 4  10 x 2  1  0
1
, or otherwise, solve the equation x x 4  4 x 3  x 2  4 x  1  0 , giving your answer in surd form.

17.

Using the substitution y  x 

18.

Determine the nature of the roots of each of the following quadratic equations.
(a)
(c)

19.

4x2  6x  1  0
2 x 2  3ax  3a 2  0

(b)

x 2  4 px  4 p 2  0

Solve the equation 5 x 4  16 x 3  42 x 2  16 x  5  0 .

2.3 Inequalities
20.

Find the set of value of x that satisfy the following inequalities.
(a)

( x  1)( x  2)  0

(b)

(2 x  1)( x  1)  0

(c)

x 2  4x  5

(d)

5 x 2  3x  2

(e)
(g)

x6
3
x2 x x

x  2 x 1

(f)
(h)

10
 19  6 x x 12
 x 1 x3 3

FHMM1014 MATHEMATICS 1

(i)
21.

(2 x  1)( x  2)
0
( x  1)

(j)

x 1

5x  3 x Solve each of the following inequalities:
(a)

(b)

x  2x  1x  3  0

(c)
22.

x2  9
2 x 3  7 x 2  17 x  10

(d)

x
1
 x  8 x 1

Find the set of values of x which satisfy each of the following inequalities.
(a)

|3x + 4| > 5

(c)

(b)

2 x  5  11

2 x 1  x  3

2.4 Partial Fractions
23.

Express the following in partial fractions.
(a)

x
( x  2)( x  3)

(b)

(c)

x2 x( x  1)( x  1)

(d)

(e)

(g)

(i)

(k)

x3
( x  1)( x  1) 2 x2  1
( x  1)( x  1)

2x  1
2

( x  3)( x  2)

(f)

(h)

(j)

x3
( x  1)( x  1) x 1
2

x( x  x  1) x 1
( x  2)( x  1) 2 x3  1 x( x  1)

2x  3
2

x (1  x)

2 x 3  3 x  23
(2 x  1)( x 2  4)

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FHMM1014 MATHEMATICS 1
24.

Express the following rational functions in the form of partial fractions.
(a)

x  13
2
x  x6

(b)

11x 2  4 x  12
(2 x  1)( x 2  4)

(c)

x3  1
(2 x  1)( x 2  4)

(d)

x2 x ( x  1)

(e)

1
( x  3 x  4) 2

(f)

3
( x  1) 2 ( x  1)

(g)

x2
( x  1)( x  1)

2

3

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FHMM1014 MATHEMATICS 1
ANSWERS
3

(b)

18

(c)

35
16

92
27

(e)

c 2  ac  b

(f)

a 3  2a 2  6

(a)

( x  1)( x  2)( x  1)

(b)

( x  2)( x 2  x  1)

(c)

( x  1)( x  1)( x 2  1)

(d)

( x  2)( x  2)( x 2  x  1)

(e)

2.

(a)

(d)

1.

(2 x  1)( x 2  1)

3.

x 2  2x  1

4.

5.5

5.

3x, −7x + 5

7.

5 2 14 68 x  ,
3
9 9
2x + 3

8.

(x + 2)(2x − 1)(3x + 2), −2, 

9.

a = 3 and b = 9

10.

(a) x + 2 (b) −2, 4, 1 ± i

11.

(i)

a  2, b  13

(ii)

x2

12.

(i)

h  3, k  2

(ii)

( x  1)(4 x  1)( x  2); x  1, x 

13.

(i)

k  14

14.

(ii)

p7

15.

(a)

x  2, x 

6.

2 1
,
3 2

x

(ii)

1
,x 1
4

1
2

x  1  2i
1
, x  2
4

x 1

(b) x  4, x 

1
1
,x 
2
3

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FHMM1014 MATHEMATICS 1
(c)
(e)

4 x  1, x   , x  3
3
3 5 x  1, x  1, x 
2

(d)
(f)

1
1
x  2, x   , x  3, x  
2
3
1
x  1, x  1, x  3, x  
3

16.

a  2 3 , b  2 3 or a  2 3 , b  2 3 , 3  2 ,  3  2

17.
18.

1  5 3  13
,
2
2
(a) 2 real and distinct roots (b) 2 real and equal roots (c) 2 complex roots

19.

−1,

20.

(a)

{x : x  1 or x  2}

(b)

(c)

{x : x  1 or x  5}

(d)

(e)

{x : 6  x  2}

(f)

(g)

{x : x  0 or 1  x  2}
1
{x :  x  1 or x  2}
2

(h)

1
{ x : 1  x  }
2
2
{x : x   or x  1}
5
2
5
{x : x  0 or  x  }
3
3
{x : 3  x  3 or x  5}

(j)

{x : x  0 or 1  x  3}

(i)

21.

1 or 5
5

(a) x  3 or x  3 (b) x < −3 or −2 < x < 1 (c)  2  x 
(d)  8  x  2 or 1  x  4

1 or x  5
2

1
1
(b)  3  x  8 (c)   x  5
3
3

22.

(a) x < −3 or x 

23.

(a)

3
2
 x3 x2

(c)



(e)

1
1
2


2( x  1) 2( x  1) ( x  1) 2

2
3
1

 x 2( x  1) 2( x  1)

(b)

2
1
 x 1 x 1

(d)

1 x  x x2  x  1

(f) 

1
1
2


9( x  2) 9( x  1) 3( x  1) 2

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FHMM1014 MATHEMATICS 1
1
1

x 1 x 1

(g)

(i)

5
5x  7

11( x  3) 11( x 2  2)

(k)

24.

1

1

1
2
 x x 1

(h)

x 1

(j)

5
5 3
 
1 x x x2

5
3x  1

2x  1 x 2  4

3
2
3
4x
(b)

 2 x2 x3
2x  1 x  4
3 3
2
3
1
9
2 x  33
(c) 
(d)   2  3 

2 x x x 1
2 342 x  1 17 x  4 x 2
1
2
1
(e) 



2
125 x  1 25 x  1 125 x  4  25 x  4 2

(a)



(f) 



3
3
3


2
4 x  1 2 x  1
4 x  1

(g) 1 

1
1

2 x  1 2 x  1

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