Grade 9 – Unit 5
Polynomials
A polynomial is the sum or difference of one or more terms
containing numbers and variables with whole number exponents.
The product of a number and one or more variables is called a term. If a
term contains no variable, it is a constant term.
Polynomials with 1, 2 or 3 terms have special names
A monomial has one term.
A binomial has two terms.
A trinomial has three terms.
For example
9xy has one term. It is a monomial.
4  3x has two terms. It is a binomial.
x 2  7 x  12 has three terms. It is a trinomial.
The degree of a polynomial is highest power of the variables.
For example
x 2  2x  3 is degree 2.
x 5  x 2  2x is degree 5.
Polynomials should always be written in descending exponent form.
From largest exponents to smallest exponents.
For example
x 3  x 5  4  x 2 should be written as x 5  x 3  x 2  4
2x 3  5x 6  2x  7 should be written as 5x 6  2x 3  2x  7
Addition and Subtraction of Polynomials
When adding or subtracting polynomials we combine only the like
terms.
If the terms contain exactly the same variables and exponents, they
are called like terms.
For Example
2x and 7x are like terms
x and y are not like terms
x 2 and 3x 2 are like terms.
x 2 and x are not like terms
5xy 2z and 2zxy 2 are like terms
3xy 2 and 2x 2 y are unlike terms
Examples
4p  7q  5q  3p
7d 2  7d  3d
 4p  3p  7q  5q
 7d 2   7  3  d
  4  3  p  7  5  q
 7d 2  10d
 p  12q
If there are brackets we need to remove them before adding or
subtracting. When removing a bracket following a negative
remember to change the signs in the brackets.
xy
2
 
 2x  3  2  4x  2xy 2

 2x
2
 
 4  x 2  3x  3
 xy 2  2x  3  2  4x  2xy 2
 2x 2  4  x 2  3x  3
 xy 2  2xy 2  2x  4x  3  2
 2x 2  x 2  3x  4  3
 3xy 2  2x  5
 x 2  3x  1
2

Practice
1.
Simplify:
(a) 9k  k
(c)
(b) 10d  6d
(d) 16r   12r 
3p  12p
(e) 10r  10r
2.
(f)
14k  14k
Simplify
(a) 4a2  3a  5a2
(c)
4x 2y  8xy 2  2x 2y
(b) 6g  f  7f  18g  3f
(d) 9  4w   15   6w  6  2w
(e) 8k   6w   14   8w   k  3w  9k  10
3
3.
Simplify:
(a)
 p  8   p  8
(b)
 4x  5   9x  14
(c)
 4km
(d)
 4x  8y    x  10y 
(e)
 4ab  5ab    2a b  ab    5ab  7 a b 
(f)
 2c  cd  5d    6c  2cd  4d 
2
 
 3m2  6  2km2  15
2
2

2
4
2
Multiplying Polynomials
To multiply monomials the numerical coefficients (numbers) are
multiplied and the variables are multiplied using the exponent rule
for multiplication.
Exponent rule: For any numbers x, m and n
x m  x n  x mn
For example
(x 3 )(x 2 )   x)  (x)  (x    x)  (x   x 5
 2mn  3mn   2   3   m  m  n  n  6  m1 1n11  6m2n2
(4x 2y 2 )(3xy 5 )  4  3  x 2  x  y 2  y 5  12x 2  1y 2  5  12x 3 y 7
To multiply a monomial with a polynomial the distributive property is
used.
Distributive Property: To multiply a factor by a sum of
numbers, multiply the factor by each number in the sum.
Then add the products.
a b  c  a  b  a  c
For example
a) 3x 2  x  4  3x 2  x  3 x 2  4  3x 3  12x 2
b)
y
2
 3y  4
  2y 
 y 2  2y   3y  2y   4  2y 
 2y 3  6y 2  8y
5
Practice
Simplify each of the following:
1.
 2x   3y 
2.
 4x   4x 
3.
  3xy   2xy 
4.
 6pq    p3q 2 
5.
8ab 8abc 
6.
 6xy 
3
6
7.
 2m n
8.
(5a 2b3 )2
9.
(5 y 2 )(4 xy 3 )
2
3
10.
(2 p)(3 p 2 q)(6 pq 2 )
11.
 3xy   2x 2y 2  
12.
3  a  b
13.
1

xy 
6

3p  p  3 
7


14.
2m2 3m3  2m2
15.
4 3  2z  z 2
16.
3 p2  2p  4
17.
4w 2w  4  w 2
18.
2r 2q r 2  q 2  1
19.
(2a)(a2  2a  1)
20.
(3pq)(p2  2pq  q2 )








8
To multiply two binomials the distributive property is used twice.
This can also be called the FOIL method.
For example
 x  4   x  1  x  x  1  4  x  1
 x 2  x  4x  4
 x2  5 x  4
 3p  1  2p  4   3 p  2p  4   1  2p  4 
 6 p2  12p  2p  4
 6p2  10 p  4
 2x  3   x  4   2x 2  8x  3x  12
 2x 2  11x  12
 2y  3x 
2
  2y  3x   2y  3x 
 4y 2  6yx  6xy  9x 2
 4y 2  6xy  6xy  9x 2
 4y 2  12xy  9x 2
To multiply a binomial by a polynomial the first term of the binomial
is multiplied by each term in the polynomial and the result is then
added to the product of the second term of the binomial with each
term of the polynomial. The result must then be simplified if
possible.
 x  2  x
2
 3x  2

 4x  2y  xy  x  y 
 x  3x  2x  2x  6x  4
 4x 2y  4x 2  4xy  2xy 2  2xy  2y 2
 x 3  x 2  8x  4
 4x 2y  2xy 2  4x 2  2y 2  6xy
3
2
2
Each term is one bracket must be multiplied with each
term in the second bracket.
9
Multiply each of the following.
1.
 x  3   3x  5 
2.
 2s  4  3s  7 
3.
 2s  4  3s  7 
4.
 3w  3  w  3 
5.
 y  4  y  4
6.
7.
9.
(3x  2)(x  5)
(2y  7)(2y  7)
8.
10.
10
(3m  2)(2m  4)
(8p  2)(8p  2)
(4x  3)2
Multiply each of the following.
11.
 3ab  2b  ab  4b
13.
 4x
15.
 y  4  y 2  4y  3 
17.
19.
2
 3y 2
  2x
2
 y2
12. 9  4w 

14.
16.
(3x  1)(5  2x  x2 )
18.
(2y  7)(2y 2  7y  1)
(5x  4)(5x  4)


(m  2) m2  3m  1
 4r  2t   r 2  2rt  t 2 
20.
11
2
(4x2  1) 2x  3y  1
Dividing Polynomials
To divide a polynomial by a monomial each term of the polynomial is
divided by the monomial. The numerical coefficients are divided and
the variables are dividing using the exponent rule for division.
Division property of exponents
For every nonzero number a and exponents m and n ,
am
 am  n
an
For example
16m2n 16 m2 n


  4m
4mn
4 m n
14p3q2
14 p3 q2



 2p2q
7 pq
7
p q
12a3 b2  6ab3  18a2b 12a3 b2 6ab3 18a2b



3ab
3ab
3ab
3ab
2
2
 4a b  2b  6a
50mn2  25m2 n 50mn2 25m2n


5mn
5mn
5 mn
 10n   5m
 10n  5m
12
Practice
1.
33x 2  11x  11
11
2.
14n2  7n  28nx
7n
3.
3cd  42cd 2
3cd
4.
8x 2y  24x 2y 2  8xy 2
8xy
5.
8mn  4mn2  4n2
4n
7.
 63k
9.
2
6.

 7k  21kn   7k 
 2x
2
 
y  x2  x2

2
2
8. (45xy  72xy  35xy)  (9xy)
(15p2q2r 2  20pqr )  ( pqr )
10.
13
(12abc  6ac  18bc)  (6)
Simplifying Polynomials
To simplify polynomials that have more than 1 operation remember
to follow the order of operations.
Brackets
Exponents
Division or Multiplication (from left to right)
Addition or Subtraction (from left to right)
Simplify each of the following
1.
2  3y  5   12
2.
7  c  4  2  c  18 
3.
14p  2  r  7 p  6r
4.
y  6y  8   5 y 2  2y  1
14


5.  4a  1  3a  2   a  2  a  2
6.
 c  4
7.
8.
4s  p  s    p  2s   p  2s 
9.
 2x  1  2x  1   2x  3 
2
5y  2y  4    y  2   4y  7 
7
10.
15
2
  2c  1  c  5 
6r  4m  5r   3m  8r  11mr 
3r
11.
a (12a2b2  4ab  1)  a
13. 2(y  3)2  3(y  1)2
12.
(m  n)2  (m  n)2
mn
14. 7a(a2  b2 )  2ab  2b(a2  b2 )
16