algebraic expressions

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
25x4
2. 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 – 3x
5.
–5x + 7
3x – 12+
4x – 3
–2x + 8
6.
3x + 1
+
x + 3
4 – x2
11. x2 – 4x + 4
x – 4
+
x + 5
–x2 + x + 2
12.
x2 – x – 6
3x + 1
+
2x + 3
9 – x2
13.
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 – x2
16.
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 – 3x
7. –5x + 7
3x – 4 + 4x – 3
–6x + 8
8.
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
5x3
3. 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. –x(x2 – x + 2)
(3x+2)(x+1)(x–1)
15. x (x + 2)
(x2 + 2)

14. Multiplication and Division of Rational Expressions
Exercise D.
1. (x + 3)(x – 1) 3. 5.
–(x + 1)(2x + 3) (2x – 3)(3x + 2)
7. 3x2 – 12x – 1 9. – 5x – 5 11. – 3x – 3
Exercise E.
3.
7(x + 1)
2(3x – 2)
5.
x + 3
1 – 2x
1. 7.
x2 + 9
9 – x2
4(x + 2)
3x – 2
34x2 – 9x – 20
5(2x – 5)(2x – 1)
9. 2(x2 + 3x + 4)
(x – 2) 2(x + 2)
11.
x2 + 3x – 3
(x2 – 9)(x + 2)
13. x2 + 8x – 12
x(x – 1)(x + 3)
15.

15. Complex Fractions
– 4x + 1
6x + 4
3. (6x-17)(x+4)
5.
14(2x+3)(x+1)
15
11
1.
Exercise F.
x2 + x + 6
7.
x2 + 3
– 2
x(h + x)
9. –4
(x - h)(x + h)
11.
Exercise G.
1. 3 – 2 3. 11 – 73 5.
3
20
(4 – 6) 13 – 14
10
7.
– 9x + 15x – 6
9.
4 – 9x
13. 1
x + h + x
(x – 4)(x+2 – 2)
11.
x – 4
15. 5
5x+5h+1 + 5x+1