Name: _____________________________
Polynomial Expressions
Date:_____ Period:____
Ms. Anderle
Polynomial Expressions:
A ________________ is a constant, a variable, or a product of constants and
variables.
Examples:
A _______________ is the sum of monomials. Each monomial is a term of the
polynomial. The same properties that are true for integers are true for polynomials:
we can use the commutative, associative, and distributive properties when working
with polynomials.
Operations with Polynomials:
When two polynomials are added, the two terms that have the same power of the
same variable are combined into a single term. Two terms that have the same variable
and exponent or are both numbers are called like terms.
Two monomials that are not like terms cannot be combined. When subtracting
polynomials, you must make sure to distribute the negative sign.
Examples:
Write the sum of difference of the given polynomials in simplest form.
1. (3y – 5) + (2y – 8)
2. x2 + 3x – 2 + 4x2 – 2x + 3
3. (4x2 – 3x – 7) + (3x2 – 2x + 3)
4. (-x2 + 5x + 8) + (x2 – 2x – 8)
5. (a2b2 – ab + 5) + (a2b2 + ab – 3)
6. (7b2 – 2b + 3) – (3b2 + 8b + 3)
7. (3 + 2b + b2) – (9 + 5b + b2)
8. (4x2 – 3x – 5) – (3x2 – 10x + 3)
9. (y2 – y – 7) + (3 – 2y + 3y2)
10. (2a4 – 5a2 – 1) + (a3 + a)
Polynomials with Rational Coefficients
Whenever a polynomial has rational (fractions) coefficients, we must find a common
denominator for the fractions.
For example: Simplify:
1 1

2 6
Examples:
Simplify each expression. Complete work on a separate sheet of paper.
3
 3

1
1
1.  a 2  a  2    a 2  a  4 
2
3
4
 8

3
 2

2
2.  x 3  2x  5    x 3  x  2 
5
4
 3

3
1
2  1
3
4
3.  y 2  y      y 2  y  
2
5  4
4
3
8
1
 5
5
1
3
4.  a 2b 2  ab  1    a 2b 2  ab  
12
4
2
3
 6
1
1  2
7
5.  b 2  2b     b 2  
2 3
8
4
 4
 2
1
2
4
6.   x 2  x  12    x 2  x  
4
3
5
 15
 5
2

1  1
1
7.  a 2  a    a 2  a  3 
8  6
12
9

4
 1
3 
8.  b 2  9b  4    b 2  b 
4 
5
 4
3
1
6  3
6
1 
9.  y 3  y     y 3  y 2  y 
4
8 4
5
16 
 12
5

5
1  3
3
10.  b  b 2     b 2  b  3 
4
2 8
12
6

1
3  3
5
1
11.  g 2  g    g 2  g  
5  8
4
2
3
 10
4  5
4
12.  b 3  b     b 3  6b 2  
7   12
9
8
5
1   4
2
5 
13.  y 4  y 2     y 4  y 3  y 
2   8
3
12 
6
7
1  4
1 
14.  b 2  2b  b    b 2  b 
4   21
6 
3
2
1
3 
1 3 5
15.  a 3  a 
   4a  a  
2
14  
8
7
7

 

16.  9 a 5  1 a 2  1 a    3 a 5  5 a 3  2 a 2 
 10
8
2  4
6
7

Multiplying Rational Expressions:
Multiplying rational expressions is the same as multiplying numerical fractions. Simply
put, we must first multiply the numerators (the tops) and then multiply the
denominators (the bottoms). Basically, we are just multiplying across the fraction.
Then, we reduce our fractions!
For Example:
s x sx
, where t  0 and y  0
 
t y ty
If possible, you may simplify before you multiply. This might make the problem easier.
However, you will get the same answer regardless.
Examples:
Multiply each rational expression. Simplify each expression. Complete all work on a
separate sheet of paper.
1.
18a 3 5b 4

20b 2 6a
2.
12n 3 24a 2b

8ab 2
6n 4
3.
13ab 2 25c 4

20bc 26a 3b
4.
3a 4
8b

2
15b 12a 2b
5.
125x 4 y 2 16n 5

20n 3
25xy 2
6.
14ab 20b 7

16b 4 21a 3b
7.
72d 5e 27d

81e 3 28d 12
8.
60x 2y
8z 2

24z 3 25xy 5
9.
63y 3z 5
9y

10
27z
28y 2
18b 4 16a
10.

3a 2 24b
100a 8b 3 16b
11.

60a 4
15b 5
63y 5
27z 9
12.

49x 4z 2 12y 13
120y 6z 24a 7b
13.

48a 6b 2 21y 4z
44 g 12 99b 3
14.

121b 21 48 g 4
64u 3 24b 15a
15.

3u
16a 7b 5
Dividing Rational Expressions:
Dividing rational expressions is the same as dividing numerical fractions. When
dividing rational expressions, you must multiply by the reciprocal. Simply put, you
keep the first fraction, change the division sign to a multiplication sign, and then flip
the second fraction. This is also known as “keep, change, flip.” Then after you “flip”
you can reduce anything that needs to be reduced, or you may wait to reduce at the
end.
s x

t y
s y

For Example:
t x
sy
tx
Now Try,
84 48

6 12
Examples:
Simplify each expression. Complete all work on a separate sheet of paper.
1.
70y 2z 7 21y 5z

40ab
28a 5
2.
45x 5y 2 36x 7 y 3

64a 12
16a 18
3.
32a 4b 24a 3y

7xy
28x 3b 2
4.
144y 4z 5 48yz 3

20b 2d
15b 6d 3
5.
81d 4e 27d 12e 5

48k 8
16k 4
6.
66z 4 132z

36b 4 12z 8b
7.
98xy 5 14x 4y 3

24b
56b 8
8.
32u 3d 40ub 2

15db
30d 8
9.
125q 5r 2
180q

3
30r
42q 9r 4
Multiplying Polynomials:
To multiply polynomials, we use the distributive property. Basically, we are
“distributing” each monomial in the first expression to each one in the second
expression. Then, combine all like terms.
Let’s Try…
 x  2 x  4 
Whenever we multiply two binomials together it is known as ______________.
Let’s try something more difficult:
x
2

 2x  3  x  9 
1 2
 2 2

 x  5x  2  x  x  4 
5
2


Examples:
Complete all work on a separate sheet of paper. Show all necessary work.
1.  x  8  x  3
2.  x  y  x  2y 
3. 2a  9 3a  1 
4. w  4 2w  7 
2

5. 9  k   k 
3

1

1
6.  r  3  2r  
2
3





7. w 2  9w  2 w  10 

10.  p  2 p 2  9 p  4






9. 2xy  3 5xy  3
11. r 2  8r  16  4  r 
12.  7r  2 12r  5r 
8. y 2  9 y 2  3y  4

13. k 2  3k  2 k 2  9k



16. y 2  8y  2 y 2  9y  2



14. 3k 2  12k  3 k  10 

15.  9yz  y  yz  2 
1
 1

17.  y  9  y  2 
2
 3

1
 2

1
18.  y 2  2y  1  y 2  y  6 
2
2
 5

1
 1

1
19.  x 2  x  2  x 2  10x 
2
3
 2


 4

1
20.  5b 2  b  2  b 2  12 
2

 5



1  1 2 4
21.  8 g 2 
g  g  g  2 
12  2
3

