Download B. Addition, Subtraction, Multiplication and Division of Polynomials

Buddhist Yip Kei Nam Memorial College
Secondary Four
Mathematics worksheet 1
Name： _____________________
P.1
Class (class no.)： _______ (
(More about Polynomials)
＜Revision＞
A. The General Polynomial in One Variable
A polynomial in x of degree n is an algebraic expression of the form
an x n  an1 x n1    a2 x 2  a1 x  a0
where a n  0, a n , a n1 ,…, a 2 , a1 , a 0 are real numbers and n is a non-negative
integer.
(i.e. the highest power of the variable x is called the degree of the polynomial)
e.g.
4x 6  x 3  2x 2  7
is a polynomial of degree 6.
Example 1
Determine whether the following are polynomials, if yes, put a ( ); otherwise, put a ().
Answers
(a)
 x 5  x 3  x  15
_______
(b)
5 x  5x 2  7
_______
(c)
sin x  cos x  7
_______
(d)
2
5
2x  1
_______
(e)
x  log x
_______
(f)
1
 4x 4
4
_______
Example 2
Given f x   3x 4  8 x 2  7 x 3  4  15x 5 .
(a)
Rewrite f x  in descending power of
(b)
write down the degree of f x  ;
(c)
write down the coefficients of x 3 and
(d)
write down the leading coefficient and the constant term of f x  ;
(e)
evaluate the value of f 2 .
B.
Addition, Subtraction, Multiplication and Division of Polynomials
x;
x;
Example 3
(2 x 3  5x 2  6 x  1)  ( x 2  7 x  4)  3x 3  6 x  6


)
Example 4
(2 x 3  x 2  5)(4 x  1)  x 3  2 x 2  8x  1

P.2

Example 5
(a) (24  2 x  7 x 2  x 4 )  ( x  3)
[By Long Division]
(b) ( x 3  5x 2  3)  ( x 2  x  3)
[By Long Division]
Division Algorithm
Dividend = quotient  divisor + remainder
Example 6
When a polynomial is divided by x  3 , the quotient is  2 x 2  11x  27 and the remainder is
80.
(a) Find the polynomial.
(b) Find the quotient and remainder when the polynomial is divided by x 2  x  2 .




C.
Equality of Polynomials
P.3
If two polynomials are equal for all values of x, then all the corresponding coefficients of
the terms are equal.
e.g.
If ax 3  bx 2  cx  d  px 3  qx 2  rx  s , then
a = p,
b = q,
c = r,
d=s
Example 7
If ( x  1)( x  1)( x  2)  Ax 3  Bx 2  Cx  D , find the values of
A, B, C and
D.
Example 8
If 15 x  12  A( x  1) 2  B( x  1)x  2  C x  2 , find the values of A, B and C.
2
Buddhist Yip Kei Nam Memorial College
Secondary Four
Mathematics worksheet 2
Name： _____________________
P.1
Class (class no.)： _______ (
)
(More about Polynomials)
＜Theorems about Polynomials＞
A. Synthetic Division
Example 1
By using the synthetic division, find the quotients and remainders of the following:
(a)
x 4  3x 2  5 x  4   x  2 ;
(b)
x 3  x 2  3x  1  x  3 ;
(c)
 2 x 4  3x 3  4 x 2  8x  2 x  3 .



B.



Remainder Theorem
When a polynomial f(x) is divided by (x  a), the remainder is equal to f(a).
b
Note: If a polynomial f(x) is divided by (ax  b), the remainder is f ( ) .
a
Example 2
Find the remainders when
x 3  4 x 2  5x  7 is divided by (a)
x  1,
(b)
2x  1 .
Example 3
When the polynomial x 2  12 x  3
value of k.
is divided by ( x  k ), the remainder is k 2  1 . Find the
Example 4
P.2
3
2
When the polynomial 2 x  ( k  1) x  3kx  5 is divided by ( x  2 ), the remainder is 1. Find
the values of
k.
Example 5
When the polynomial f x   x 3  ax 2  4 x  b is divided by x  3 , the remainder is 40; and
f x  is divisible by x  5 . Find the values of a and b.
Example 6
Given that f x   ax 2  3x  2
they are divided by x  1 .
(a)
(b)
and
g x   x 2  x  1
have the same remainders when
Find the value of a.
Find the value of x when f x   g x  .
Example 7
Given that f x   2 x 3  7 x 2  ax  5
Find the values of a and b.
is divisible by
2 x  1
and the quotient is x 2  bx  5 .
C.
Factor Theorem
P.3
If f(x) is a polynomial and f(a) = 0, then (x  a) is a factor of the polynomial f(x).
Note : If (x – a) is a factor of the polynomial
Example 8
If (x  1) is a factor of
Example 9
Prove that x  5
f(x), then f(a) = 0.
f ( x)  x 4  ax 3  4 x  7 , find the value of a.
is a factor of
f ( x ) = x 3  8 x 2  17 x  10 . Hence, factorize f ( x )
completely.
Example 10
Let g x  1  x 3  3x 2  10 x  24 .
(a) Find g x .
(b) Show that (x + 1) is a factor of g x .
(c) Hence, factorize g x completely.
D. Applications of the Factor Theorem
P.4
Method of Factorization
(i)
Factorization by taking out the common factors:
(ii)
Factorization by the cross method:
ab  ac  a (b  c)
6 x 2  x  1  (2 x  1)(3x  1)
(iii) Factorization by using some identities:
(1)
(2)
(3)
(4)
Perfect squares:
Difference of two squares:
Sum of two cubes:
Difference of two cubes:
Example 11
Given x 3  2 x 2  mx  6
(a)
(b)
Find the value of
Hence, factorize
x  2 x 2  mx  6
Example 12
Factorize x 3  x 2  49 x  49 .
Hence, solve x 3  x 2  49 x  49  0 .
(a  b) 2  a 2  2ab  b 2
a 3  b 3  (a  b)(a 2  ab  b 2 )
a 3  b 3  (a  b)(a 2  ab  b 2 )
m.
3
(a  b) 2  a 2  2ab  b 2
a 2  b 2  (a  b)( a  b)
is divisible by x  1 .
Solving Cubic Equations
(a)
(b)
(a)
(b)
completely.