This document provides examples and explanations of operations and concepts involving polynomial and rational expressions. It begins with examples of factoring polynomials and using the factored form to evaluate expressions. It then covers topics such as combining like terms in rational expressions, multiplying and dividing rational expressions using factoring, simplifying complex fractions, and rationalizing denominators involving radicals. The document aims to demonstrate techniques for working with polynomials and rational expressions through step-by-step examples and explanations of related concepts.

1. Polynomial Expressions
Following are examples of operations with
polynomials and rational expressions.
Example A. Expand and simplify.
(2x – 5)(x +3) – [(3x – 4)(x + 5)]
= 2x2 + x – 15 – [3x2 + 11x – 20]
= 2x2 + x – 15 – 3x2 – 11x + 20
= –x2 – 10x + 5
Insert [ ] and exp and
remove [ ], change signs.
(2x – 5)(x +3) – (3x – 4)(x + 5)
= (2x – 5)(x +3) + (–3x + 4)(x + 5)
= …
Or distribute the minus sign and
change it to an addition problem:
Example B. Factor 64x3 + 125
64x3 + 125
= (4x)3 + (5)3
= (4x + 5)((4x)2 – (4x)(5) +(5)2)
= (4x + 5)(16x2 – 20x + 25)
A3 B3 = (A B)(A2 AB + B2)+– +–+–

2. Example D. Determine whether the outcome is + or –
for x2 – 2x – 3 if x = –3/2.
x2 – 2x – 3 = (x – 3)(x + 1). Hence for x = –3/2,
we get (–3/2 – 3)(–3/2 + 1) which is (–)(–) = + .
It's easier to determine the sign of an output, when
evaluating an expression, using the factored form.
We write rational expressions in the factored form
in order to reduce and multiply/divide them.
Example F. Reduce 1 – x2
x2 – 3x+ 2
x2 – 3x+ 2 =
(1 – x)(1 + x)
(x – 1)(x – 2)
= –(x + 1)
(x – 2)
factor
1 – x2
Polynomial Expressions

3. Rational Expressions
Multiplication Rule:
P
Q
R
S
* = P*R
Q*S
Division Rule:
P
Q
R
S
÷ = P*S
Q*R
Reciprocate
Example G. Simplify (2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
(2x – 6)
(y + 3) ÷
(y2 + 2y – 3)
(9 – x2)
=
(2x – 6)
(y + 3)
(y2 + 2y – 3)
(9 – x2)*
=
2(x – 3)
(y + 3)
(y + 3)(y – 1)
(3 – x)(3 + x)*
–1 1
=
–2(y – 1)
(x + 3)
Multiplication and division of
rational expressions are reduction problems.
We factor and look for common factors to cancel.

4. –
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
y2 + y – 2 = (y – 1)(y + 2) y2 + 2y – 3 = (y – 1)(y + 3)
Hence the LCD = (y – 1)(y + 2)(y + 3).
Multiplying LCD/LCD (= 1) to the problem, cancel each
denominator, expand the numerators then simplify.
–
(y – 1)(y + 2)
2y – 1 y – 3[ ](y – 1)(y + 2)(y + 3)
(2y – 1)(y + 3) – (y – 3)(y + 2) = y2 + 6y + 3
So –
(y2 + 2y – 3)(y2 + y – 2)
2y – 1 y – 3
=
y2 + 6y + 3
(y – 1)(y + 2)(y + 3)
(y + 3) (y + 2)
Example I. Combine
LCD
LCD
(y – 1)(y + 3)
Build the LCD.
To combine rational expressions (F ± G),
multiple (F ± G)* LCD / LCD, expand (F ± G)* LCD
and simplify (F ± G)(LCD) / LCD.
Rational Expressions

5. Example K. Simplify
–
(x – h)
1
A complex fraction is a fraction of fractions.
To simplify a complex fraction, use the LCD to clear
all the denominators of all the fractioned terms.
(x + h)
1
2h
Multiply the top and bottom by (x – h)(x + h) to reduce the
expression in the numerators to polynomials.
–
(x – h)
1
(x + h)
1
2h
=
–
(x – h)
1
(x + h)
1
2h
(x + h)(x – h)[ ]
(x + h)(x – h)*
=
–(x + h) (x – h)
2h(x + h)(x – h)
=
2h
2h(x + h)(x – h)
=
1
(x + h)(x – h)
Rational Expressions

6. To rationalize radicals in expressions we often use
the formula (x – y)(x + y) = x2 – y2.
(x + y) and (x – y) are called conjugates.
Rationalize Radicals
h
x + h – x
= h
(x + h – x) (x + h + x)
(x + h + x)
*
=
h
(x + h)2 – (x)2
(x + h + x)
=
h
h
(x + h + x)
=
1
x + h + x
Example K: Rationalize the numerator h
x + h – x
(x + h) – (x) = h

7. Exercise A. Factor each expression then use the factored
form to evaluate the given input values. No calculator.
Applications of Factoring
1. x2 – 3x – 4, x = –2, 3, 5 2. x2 – 2x – 15, x = –1, 4, 7
3. x2 – x – 2, x = ½ ,–2, –½ 4. x3 – 2x2, x = –2, 2, 4
5. x4 – 3x2, x = –1, 1, 5 6. x3 – 4x2 – 5x, x = –4, 2, 6
B. Determine if the output is positive or negative using the
factored form.
7.
x2 – 4
x + 4
8. x3 – 2x2
x2 – 2x + 1
, x = –3, 1, 5 , x = –0.1, 1/2, 5
4.
x2 – 4
x + 4 5. x2 + 2x – 3
x2 + x
6. x3 – 2x2
x2 – 2x + 1
, x = –3.1, 1.9 , x = –0.1, 0.9, 1.05
, x = –0.1, 0.99, 1.01
1. x2 – 3x – 4, x = –2½, –2/3, 2½, 5¼
2. –x2 + 2x + 8, x = –2½, –2/3, 2½, 5¼
3. x3 – 2x2 – 8x, x = –4½, –3/4, ¼, 6¼,

8. C. Simplify. Do not expand the results.
Multiplication and Division of Rational Expressions
1. 10x *
2
5x3
15x
4
*
16
25x42. 3.12x6 *
5
6x14
4. 75x
49
*
42
25x3
5. 2x – 4
2x + 4
5x + 10
3x – 6
6.
x + 4
–x – 4
4 – x
x – 4
7. 3x – 9
15x – 5
3 – x
5 – 15x
8. 42 – 6x
–2x + 14
4 – 2x
–7x + 14
*
*
*
*
9.
(x2 – x – 2 )
(x2 – 1) (x2 + 2x + 1)
(x2 + x )
* 10.
(x2 + 5x – 6 )
(x2 + 5x + 6) (x2 – 5x – 6 )
(x2 – 5x + 6)
*
11. (x2 – 3x – 4 )
(x2 – 1) (x2 – 2x – 8)
(x2 – 3x + 2)
*
12. (– x2 + 6 – x )
(x2 + 5x + 6) (x2 – x – 12)
(6 – x2 – x)
*
13.(3x2 – x – 2)
(x2 – x + 2) (3x2 + 4x + 1)
(–x – 3x2)
14. (x + 1 – 6x2)
(–x2 – 4)
(2x2 + x – 1 )
(x2 – 5x – 6)
15. (x3 – 4x)
(–x2 + 4x – 4)
(x2 + 2)
(–x + 2)
16. (–x3 + 9x ) (x2 + 6x + 9)
(x2 + 3x) (–3x2 – 9x)
÷
÷
÷
÷

9. Multiplication and Division of Rational Expressions
D. Multiply, expand and simplify the results.
1. x + 3
x + 1
(x2 – 1) 2. x – 3
x – 2
(x2 – 4) 3. 2x + 3
1 – x
(x2 – 1)
4. 3 – 2x
x + 2
(x + 2)(x +1) 5.
3 – 2x
2x – 1
(3x + 2)(1 – 2x)
6. x – 2
x – 3
( x + 1
x + 3)( x – 3)(x + 3)
7. 2x – 1
x + 2
( – x + 2
2x – 3 )( 2x – 3)(x + 2)
+
8.
x – 2
x – 3
( x + 1
x + 3
) ( x – 3)(x + 3)–
9.
x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 3)(x + 1)
10.
x + 3
x2 – 4
( – 2x + 1
x2 + x – 2
) ( x – 2)(x + 2)(x – 1)
11.
x – 1
x2 – x – 6
( –
x + 1
x2 – 2x – 3
) ( x – 3)(x + 2)(x + 1)

10. E. Combine and simplify the answers.
–3
x – 3
+ 2x
–6 – 2x
3. 2x – 3
x – 3
– 5x + 4
5 – 15x
4.
3x + 1
6x – 4
– 2x + 3
2 – 3x5.
–5x + 7
3x – 12+
4x – 3
–2x + 86.
3x + 1
+
x + 3
4 – x211. x2 – 4x + 4
x – 4
+
x + 5
–x2 + x + 2
12.
x2 – x – 6
3x + 1
+
2x + 3
9 – x213.
x2 – x – 6
3x – 4
–
2x + 5
x2 – x – 6
14.
–x2 + 5x + 6
3x + 4
+
2x – 3
–x2 – 2x + 3
15.
x2 – x
5x – 4
–
3x – 5
1 – x216.
x2 + 2x – 3
–3
2x – 1
+ 2x
2 – 4x
1.
2x – 3
x – 2
+
3x + 4
5 – 10x
2.
3x + 1
2x – 5
– 2x + 3
5 – 10x
9.
–3x + 2
3x – 12
+
7x – 2
–2x + 8
10.
3x + 5
3x –2
– x + 3
2 – 3x7. –5x + 7
3x – 4 + 4x – 3
–6x + 88.
Addition and Subtraction of Fractions

11. Complex Fractions
1
2x + 1
– 2
3 –
1
2x + 1
3.
–2
2x + 1
–
+
3
x + 4
4.
1
x + 4
2
2x + 1
4
2x + 3
–
+
3
x + 4
5.
3
3x – 2
5
3x – 2
–5
2x + 5
–
+ 3
–x + 4
6.
2
2x – 3
6
2x – 3
2
3
+ 2
2 –
–
1
6
2
3
1
2+
1.
1
2
– +
5
6
2
3
1
4
–
2.
3
4
3
2
+
F. Combine and simplify the answers.
7.
2
x – 1
–
+
3
x + 3
x
x + 3
x
x – 1
8.
3
x + 2
–
+
3
x + 2
x
x – 2
x
x – 2
9.
2
x + h
–
2
x
h
10.
3
x – h
–
3
x
h
11.
2
x + h – 2
x – h
h
12.
3
x + h
–
h
3
x – h

12. G. Rationalize the denominator.
1.
1 – 3
1 + 3
2.
5 + 2
3 – 2
3.
1 – 33
2 + 3
4.
1 – 53
4 + 23
5.
32 – 33
22 – 43
6.
25 + 22
34 – 32
7.
42 – 37
22 – 27
8.
x + 3
x – 3
9. 3x – 3
3x + 2
10. x – 2
x + 2 + 2
11. x – 4
x – 3 – 1
Algebra of Radicals

13. (Answers to odd problems) Exercise A.
Applications of Factoring
1. (x + 1)(x – 4), 6, – 4, 6 3. (x + 1)(x – 2), – 9/4, 4, – 5/4
Exercise B.
1. positive, negative, negative, positive
3. negative, positive, negative, positive
5. x2(x2 – 3), – 2, –2, 550 7. , –3/5, 7/3
5. positive, negative, positive
Exercise C.
1. 4
x2
12
5x33. 5. 7. 3(x – 3)2
25(3x – 1) 2
5
3
(x + 2)(x – 2)
x+4
9. x (x – 2)
x2 – 1
11. x – 2
x + 2
13. x3 + x + 2
x (–3 x2 + x + 2)
15. x (x + 2)(x2 + 2)
(x – 2) 2