Basic Concepts of Algebra

2. ALGEBRAIC EXPRESSIONS

2    ALGEBRAIC EXPRESSIONS
2.1 USE OF SYMBOLS (Literal and numerical):
    In algebra letters are frequently used to represent numbers. This practice enables us to calculate the numerical value of certain quantities or terms at the outset of a problem. A letter that represents a number is termed a literal number. When two or more numbers are multiplied together the numbers are called factors. (Also two or more letters multiplied together or two or more mixed numbers and letters multiplied together the members are called factors). Thus, when multiplying in algebra we refer either to numerical factors (digits) or literal factors (letters).

EXAMPLE 2.1A
                  Look at the factors in the number 10. Now 10 = 1 x 10, 2 x 5. Therefore 1, 2, 5 and 10 are numerical factors.
EXAMPLE 2.1B
                Look at the factors in the term  5 x y. Now  5xy = 5 ' x  'y . Therefore 5 is the numerical factor and x,y are the literal factors.

2.2  Some important definitions
1.  A variable is a letter or symbol used to represent some unknown quantity. For example, x is often used as a variable.
        The numerical coefficient of a variable is the number that multiplies the variable. In the term 5x, the coefficient of a x is 5.
Examples: 2x2 ... The numerical coefficient of x2 is 2
                  -3x .. The numerical coefficient of x is -3
                      x... The numerical coefficient of x is 1
        It is wise to note that to show multiplication of variables, the “times” sign is usually omitted; since 5 x x;  5∙x and 5x mean 5 times x.
2. ... An algebraic expression is a collection of numbers and variables. Thus, 2x+y is an algebraic expression which means ‘’two times x plus y’’... 5 x2 is an algebraic expression which means “5 times x times x”.

3. ... A term is an algebraic expression with number(s) and variable(s) connected only by the operations of multiplication and division.
Examples: 5x,5xy,3aby, -5a2, 4/x,x2y2/c.
4.... Like terms contain the same variable or variables and differ only in their numerical coefficients. In the expression 2x+5x the terms are like that differ only with respect to their coefficients 2 and 5. The terms 2x+5x can be combined to form a single term; thus, 2x+5x=7x ← single term.

5... A monomial is an algebraic expression with one, and only one, term.
Examples: 3bxy,2a2b,2(3 x y),abc/2.  A binomial is an algebraic expression with two, and only two, terms separated by either a plus or a minus sign.
Examples: 2x+5y,2x2-3y,ab+2a2, wxy+3p.

7...A trinomial is an algebraic expression with three, and only three, terms separated by either plus or minus signs or plus and minus signs.
Examples: x2+ 5x+6,y2-3y-2,2a+3b-c,3ax^2-2b+8.

8...A polynomial is an algebraic expression with more than one term. A polynomial is a general name for binomials as well as trinomials.
Examples: 2a+3b,abc-2cd+3de,x2-4x-5,2a-3b-4c-2x+4ay

9...An exponent is the index to which a base is raised. Thus, the expression x^n is read, the base x raised to the nth power. x is the base but n may be referred to as the index, the power or more commonly the exponent.
Examples:
                  a^2= a ' a
                  x^3= x ' x ' x
                  2^3 y^2= 2 ' 2 ' 2 ' y ' y

NOTE: The laws of exponents will be dealt with later.

10. The degree of a polynomial is the highest value of an exponent in one of its terms.
Examples:
                  4x^2+ 3x+1 → 2nd degree
                  3x^4+ 2x^3+ 4x^2– 8 → 4th degree



NOTE:  For POLYNOMIALS there are restrictions:
1.  The exponent of letters must be positive integers.
2.  Variables (letters) should NOT be in the denominator
3.  Variables should Not be under a radical sign
Examples: (x-3)/5x,7/x-2/2x2,and 5√x -5
Are merely algebraic expressions not POLYOMIALS.

2.3 USE OF VARIABLES IN ALGEBRIAC EXPRESSIONS:
Lets analyze the following expressions.
Statements                                                                                                                                          Symbols
∙ Three times a number                                                                                                                    3 ' x or 3x
∙ 5 more than a number                                                                                                                  x + 5
∙ 4 less than a number                                                                                                                      x- 4
∙ One– seventh of a number                                                                                                          1/7 x or x/7
∙ A number squared                                                                                                                          x2
∙The product of two numbers                                                                                                        xy                     
∙ The quotient of x and 2                                                                                                                  x/2
  Recall the definition of a variable 2.2   






      EXERCISE 2.3
                Translate the following statements into algebraic expressions.
      Use the variable x  to represent the unknown unless otherwise noted:

1.  Twice a number, increased by 5.

2.  The difference between a number and 5.

3.  7 less than twice a certain number.

4.  The product of a certain number and 7.

5.  The quotient of a certain number and 4.

6.  8 minus four times a certain number.

7.  Twice a number increased by 12.

8.  The quotient of 5 minus a number and 6.

9.  10 minus the sum of 12 and a number.

10. The product of 3 and the difference of x and y.                                                                       




2.4 EVALUATING ALGEBRAIC EXPRESSIONS:
          Finding the value of an expression when the variables are known is called evaluating an expression.
Finding the value of 2x+3y when〖 x〗^1 =5 and y=2
Step 1      Substitute the given values → 2(5)+ 3(2) for variables.
Step 2      Follow the order of operation: MULTIPLY →2'5= 10 and 3'2= 6
                                                                              ADD      → 10+6=16








      EXERCISE 2.4
      Evaluate the following:

1.  x^2- 2y  if x=5 and y=3

2.  x^2+ xy if x=2 and y=^- 2

3.  (a-b)(a+b)  if a=5 and b= -5

4.  4(x-2)+2y if x=^- 7and y=10

5.  x^2-y^2 if x=^- 2 and y=1

6.  -3x(x^2+ 2)when x=5

7.  A= L ' W when L=15 and W=12

8.  A=πr^2 when π=22/7 and r=14

9.  C=πD when π=22/7  and D=7^2

10. I=PTR when P=500,T=5 and R=0.05






2.5 ADDING ALGEBRAIC EXPRESSIONS:
      When adding algebraic expressions, we combine like terms.
Example: Simplify    3x+2y-5τ+2x+6τ
Step 1 Combine like terms          3x+2x=5x   
                                                          6τ+ -5τ=τ 
Step 2 Write down the sum
→ 5x+ τ+2y
     





EXERCISE 2.5                 
Simplify the expression:

1.  5x+4y+3x-5y

2.  7x+3y-4x+2y

3.  8a+6b+5c

4.  2x^2-2x+3x^2-6+4x+5

5.  -5a+17b+10-7b+6a-7

6.  7a-8b+9a

7.  9x-6+8x^2+17

8.  15-5n+11n-2

9.    7+15n-12-4n

10. 4m-3+9n^2+3m-4n^2






2.6      MULTIPLYING ALGEBRAIC EXPRESSIONS:
            Let us begin with single-term expressions.
NOTE: To multiply algebraic expressions; first multiply numerical factors, then multiply the variables. To multiply a power of x by another power of x, you add the exponents. (We will discuss exponents in details, later).
Examples:
                (2x^3 )(5x^2 )               
Multiply numerical factors → 2 ' 5 = 10
Multiply variables x^3' x^2  → x∙x∙x  ∙  x∙x →x^5
Therefore 10x^5 is the product.
→ (2x3) (5x2) = (2)(5)(x3)(x2)=10x3+2
                                                                                          = 10x5
∙ To multiply more complex expressions, use the distributive law (as discussed in 1.3)
Examples:
1. 2(x+3y)            ⇒2∙x+2∙3y⇒2x+6y
2. 3x2(2+y)            ⇒3x^2∙2+3x2 ∙y⇒6x2+3x2y
3. ab(a+b)            ⇒ab∙a+ab∙b⇒a2b+ab2
4. a2b(2a-5b)      ⇒a2b∙2a-a2b∙5b⇒2a3b-5a2b2
5. (x+2)(2x-3)⇒(x∙2x)-(x∙3)+(2∙2x)-(2∙3)
                                    ⇒2x2-3x+4x-6
                                                        ⇒2x2+x-6
∙ To multiply a binomial (2-term expression)by a binomial use the FOIL (First, Outer, Inner, Last) method. If you multiply the terms in this order, you can be sure that you have multiplied every possible combination of terms.
(x+5)(x+3)
Multiply first terms: F→(x+5)(x+3)→x^2
Multiply outer terms: O→ (x+5)(x+3)→3x
Multiply inner terms: I→(x+5)(x+3)→5x
Multiply last terms: L →(x+5)(x+3)→15
Simplify the expression. x^2+3x+5x+15
                                          →x^2+8x+15
∴ The product of x+5 times x+3 is x^2+ 8x+15.
(x+5)(x+3)=x^2+8x+15 (trinomial) binomial times binomial = trinomial
Examples:
1                (x+7)(x-3)
                F    (x+7)(x-3)                    x∙(x)=x^2
                O    (x+7)(x-3)                    x∙(-3)=-3x
              I      (x+7)(x-3)                    7∙(x)=7x
              L      (x+7)(x-3)                    7∙(-3)=-21






Simplify the expression.
〖Thus,  x〗^2-3x+7x-21→x^2+4x-21
∴ (x+7)(x-3)↔x^2+4x-21
Multiplication →
                          ← Factorization
2. (x-8)(x-5)
              F    (x-8)(x-5)                        x∙(x)=x^2
                  O    (x-8)(x-5)                        x∙(-5)=-5x
          I    (x-8)(x-5)                      -8∙(x)=-8x
        L    (x-8)(x-5)                      -8∙(-5)=40
Simplify the expression.
    〖Thus,x〗^2-5x-8x+40→x^2-13x+40
∴ (x-8)(x-5)        ↔x^2-13x+40







EXERCISE 2.6  Multiply the expressions:

1.  (2x^2 )(3x^4 )

2.  5(3x-7)

3.  2x(5x^3+2)

4.  5x(2x+7)

5.  ab(a^2+3b^2 )

6.  (a+b)(2a+b)

7.  mn(m^2+2n^2)

8.  12(2p-4q)

9.  2ab^3 (3a-4b^2)

10. (x+2)(x+5)

11. (x-1)(x+5)

12. (x+2)(x-2)

13. (x+6)(x+3)

14. (x-7)(x-5)

15. (x+10)(x-12)







2.7  DIVISION OF ALGEBRAIC EXPRESSIONS:
        To divide algebraic expressions; first divide the numerical factors, then divide the variables. To divide a power of x by another power of x, subtract the exponents. (We will discuss exponents in details, later).
Examples: 1. 10x^5'2x^2  ⇒  (10x^5)/(2x^2 )
⇒ 10xxxxx/2xx  ⇒5xxx⇒5x^3
               
But
      (10x^5)/(2x^2 )=10/2 x^(5-2)=5x^3


2.  8x^4'2x ⇒ (8x^4)/2x 
                                     
                                        ⇒ 8xxxx/2x

                          ⇒ 4xxx ⇒4x^3 
                     
But (8x^4)/2x=8/2 x^(4-1)=4x^3

3.  (8x^3+4x^2-2x)/2x    Each term must be divided by 2x.

Thus:  〖8x/2〗^(3-1)+〖4x/2〗^(2-1)-〖2x/2〗^(1-1)
                            Note: x'=1
                                        x^(1-1)=x'=1
                                            (Will be discussed in indices)
⇒ 4x^2+2x-1






    EXERCISE 2.7
    Divide the expressions:

1. (18a^4)/(6a^3 )   

2. 〖15s〗^5/〖15s〗^5

3. 12b^7 /16b^5

4. 27x^4 y^5 z^2 / 9p^2 y^3 z^2

5. (15a^3+20a^2-25a)/5a



2.8 GROUPING SYMBOLS IN ALGEBRAIC EXPRESSIONS:
      There are three common types of grouping symbols namely:
Parentheses: (  ) as in 2 (3x+y)
Brackets [ ] as in 5 [ 2-(3x+y)]
Braces {  } as in t-{5+[y-(3+x) ]}
      When more than one set of grouping symbols is used, remove one set at a time, beginning with the innermost set. The symbols: (  ),[  ], and {  } indicate that the quantities enclosed within them are considered to be a single unit with respect to anything outside the grouping symbol. For example, in the expression 7x+(5m-n)+2y, the terms 5m and-n in parentheses are considered as a single unit separate from the terms outside the parentheses.
Rule1: When removing parentheses preceded by a plus sign, do not change signs of the enclosed terms.
Examples:
              1. 2(x+y)⇒(2∙x)+(2∙y)⇒2x+2y
              2. 4b+(3b-5)⇒4b+3b-5⇒7b-5
Rule2: When removing parentheses preceded by a minus sign, drop the minus sign and parentheses and change the sign of each enclosed term.



Examples:
              1. -3(x-5)⇒-3x+15
      Note: -3∙x=-3x and -3∙-5=+15  Thus, -3x+15   
              2. (2ab-3cd)-(-3ab+2cd)=2ab-3cd+3ab-2cd=5ab-5cd





Rule3: When more than one grouping symbols is used, remove one at a time, beginning with the innermost set.
Example:
                2x-{3-[7+(4x-5) ] }
Solution:
      Remove parentheses  2x-{3-[7+4x-5] }
      Remove brackets        2x-{3-7-4x+5}
      Remove braces            2x-3+7+4x-5
      Combine like terms    2x+4x+7-5-3
                                          ⇒6x-1





EXERCISE 2.8  Simplify:

1.  2(4ab+3c)+3(a-2ab)-4(c-2ab)

2.  x-[5-2(x-1)]

3.  2(m-1)-3(2-3m)-(m+1)

4.  6+{2x-[3y+(4-5x-5y) ] }

5.  2x-{5-[3b-(5x+7-b) ]+2}

6.  a^2 b+{3-[4a+(2a^2 b-7)]}+a

7.  (5y-2a+1)+2(y-3a-7)

8.    7x-5[2x+(3-x) ]

9.    7x-2y+5+(2x+5y-4)

10. (3m+5mn+2n)+(4-3mn-2m)







2.9  SUMMARY EXERCISE for Ch 2.
        Simplify:

1.  4ab-6bc+3bc-2ab=

2.  a^2-y^2+4-3y^2=

3.  abc+bac+acb=

4.  3a^2+ab+c+2c-4a^2-ab=

5.  x^3+2x+x^2-4+x-2x^2=

6.  (-7x^2 y)-(2x^2 y)=

7.  (-2z^2 )-(-5z^2 )=

8.  (15ab)-(4ab)=

9.  (-3xy)-(xy)+(4xy)-(-5xy)=

10. (-2abc)-(-3abc)-(4abc)-(-abc)=

11. (x^2-3x^3+4x)-(x^3-2x+y-3x^2 )=

12. (5x+4y+8)-(3x-2y+6)

13. (x^2+y^3 )-(2y^2-x^2 )+(w)=

14. 2a-{4-[3b-(5a+7-b) ]+2}=

15. (12a^4-8a^3-4a^2)/(2a^2 )=

16. 5x-4+12t^2+4x-3t^2=

17. [(x+5)(x+3) ]-(7x+15)-x^2=

18. (3x^2+4x-6)+(2x^2-2x+5)=

19. (2x+4)(3x+10)

20. (x-5)(3x+2)(8x)