Required knowledge and skills: algebraic skills
Overview
1. operations with algebraic expressions
a. substituting values for variables
b. addition, subtraction, multiplication, division, exponentiation and extraction of roots
c. brackets
d. distributive properties
e. special products: (a  b) 2  a 2  2ab  b 2 , (a  b) 2  a 2  2ab  b 2 , (a  b)(a  b)  a 2  b 2
f. simplification of fractions
g. multiplication and division of fractions
h. common denominator of fractions
i. addition and subtraction of fractions
2. factoring polynomials:
a. common factor
b. grouping terms
c. special products
d. factoring polynomials of the form x 2  (a  b) x  ab , where a and b are simple integer numbers
3. powers
a. powers with natural, negative or fractional exponents, radicals
b. rules for calculations with powers and radicals:
i. product/quotient of powers with same base
ii. power of a power
iii. product/quotient of powers with same exponent
iv. power of a product/quotient
4. equations (only the purely mathematical aspects, no applications or word problems)
a. operations that either do or don’t guarantee equivalence of equations
b. linear equations
c. quadratic equations
d. simple higher order polynomial equations by using factoring and the zero-product property
e. simple rational equations
f. power equations
g. simple radical equations
5. linear inequalities
6. systems of two linear equations
Examples
Operations with algebraic expressions
Example 1. Evaluate  2(a 2  b)  3ac if a = 2, b = –1, c = –3.
1
Example 2. Work out: ( x 2  3x)(5x  1)  4 x( x 3  x 2  3x  1) .
Example 3. Work out: (3x 2  5) 2 .
1 1

Example 4. Simplify: 3 4 .
1 1

3 4
Example 5. Simplify:
3x  2
x
.

( x  1)( x  3) ( x  1)( x  2)
Example 6. Simplify
x2  y2
1.
x2
x2  y2
xy
Example 7. Simplify
.
x
y2
Factoring polynomials:
Example 8. Factor x 4  2 x 3  16 x 2  32 x .
Example 9. Factor 16 x 4  24 x 2 y 3  9 y 6 .
Example 10. Factor z 2  12 x  20 .
Powers

Example 11. Write 64
Example 12. Calculate
Example 13. Simplify
2
3
4
with positive integer exponents and roots only and calculate without calculator.
81 without calculator.
x
 
Example 14. Simplify x 5
1
x2
7
5
.
Equations (only the purely mathematical aspects, no applications or word problems)
Example 15. Solve for t: 2t  7  3(t  2) .
Example 16. Solve for x:  x 2  3 x  10  0 .
Example 17. Solve for s:  s 3  4 s 2  s  4  0 .
Example 18. Solve for x:
1
1
3

 .
x 1 x 1 4
3
Example 19. Solve for z: 4 z 4  32 .
Example 20. Solve for x: 2 x  2  11  3 .
2
Linear inequalities
Example 21. Solve for x: 2 x  7  3( x  2) .
Systems of two linear equations
2 x  2 y  1  0
Example 22. Solve the system 
x  4 y  1  0
Solutions to the examples
Example 1. –24
Example 2.  4 x 4  9 x 3  2 x 2  x
Example 3. 9 x 4  30 x 2  25
Example 4.
1
7
Example 5.
4x 2  x  4
( x  1)( x  2)( x  3)
y2
x2
Example 6. 
y( x 2  y 2 )
x2
Example 7.
Example 8. x( x  2)( x  4)( x  4)

Example 9. 4 x 2  3 y 3

2
Example 10. ( z  2)( z  10)
Example 11.
1
3
64
2

1
16
Example 12. 3
Example 13.
1
x x
Example 14. x 7
Example 15. t = –13
Example 16. x = –2 or x = 5
Example 17. s  1 , s  1 or s  4
Example 18. x  3 or x  
1
3
Example 19. z  16
Example 20. no solutions
3
Example 21. x  13
Example 22. x  1 and y 
1
2
4