1.1. Introduction to Algebra

Introduction

Review of basic algebraic terms:

Algebraic term Description Example
 Algebraic expression A mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, , ×, ÷, etc.).

3x – 4

5a2 b + 3

12y3 + 7y2 5y + \frac{2}{3}

Constant A number on its own. 2y + 5        constant:        5
Coefficient The number in front of a variable.

-9x2    coefficient:    -9

 x    coefficient:    1

(x = 1 · x)

 

Term

A term can be a constant, a variable, or the product of a number and variable. (Terms are separated by a plus or minus sign.)

     2x3 + 7x2 9y 8

Terms: 2x3,   7x2,   – 9y,   –8

Like terms The terms that have the same variables and exponents (differ only in their coefficients).

2x      and     -7x

-4y2      and      9y2

0.5pq2      and      \frac{2}{3}pq2

 

Polynomial: an algebraic expression that contains one or more terms.                                     

Example:    7x ,       5ax – 9b ,       6x2 5x + \frac{2}{3} ,       7a2 + 8b + ab – 5

 

There are special names for polynomials that have one, two, or three terms:

  • Monomial: an algebraic expression that contains only one term.

Example:       9x ,       4xy2 ,       0.8mn2  ,       \frac{1}{3}a2b

  • Binomial: an algebraic expression that contains two terms.

Example:          7x + 9 ,        9t2 – 2t ,         0.3y + \frac{1}{3}

  • Trinomial: an algebraic expression that contains three terms.

Example:          ax2 + bx + c ,       4qp2 + 3q + 5

 

Combining Terms

Like terms: terms that have the same variables and exponents (the coefficients can be different).

Examples:

Example Like or unlike terms
7y   and   -9y Like terms
6a2,   -32a2,   and   –a2 Like terms
0.3 x2y   and   -48x2y Like terms
\frac{-2}{7}u2v3   and   u2v3 Like terms
-8y   and   78x Unlike terms
6m3   and   -9m2 Unlike terms
-9u3w2   and   -9w3u2 Unlike terms

 

Combine like terms: add or subtract their coefficients and keep the same variables and exponents.

Note: unlike terms cannot be combined.

Example 1.1.1

1) 3a + 7b – 9a + 15b = (3a – 9a) + (7b + 15b) Regroup like terms.
= -6a + 22b Combine like terms.
2) 2y2 – 4x + 3x – 5y2 = (2y2 – 5y2) + (-4x + 3x) Regroup like terms.
= -3y2 – 1x Combine like terms.
= -3y2 – x
3) 8xy2 – x2y + 4x2y – 6xy2
= 8xy2 x2y + 4x2y – 6xy2 Or underline like terms without regrouping.
= 2xy2 + 3x2y Combine like terms.
4) 2(2m + 3n) + 3(m – 4n) = 4m + 6n + 3m – 12n Distributive property.
= 7m – 6n Combine like terms.

 

Removing Parentheses

If the sign preceding the parentheses is positive (+), do not change the sign of terms inside the parentheses, just remove the parentheses.

Example:     (x 5) = x 5

If the sign preceding the parentheses is negative (-), remove the parentheses and the negative sign (in front of parentheses), and change the sign of each term inside the parentheses.

Example:    – (x 7) =x + 7

 

Remove parentheses:  

Algebraic expression Remove parentheses Example
  (ax + b) ax + b (5x + 2) = 5x + 2
  (ax b) ax b (9y – 4) = 9y – 4
– (ax + b) -ax b – (\frac{3}{4}x + 7) = \frac{3}{4}x 7
– (ax b) -ax + b – (0.5b – 2.4) = -0.5b + 2.4

 

Example 1.1.2

1)  9x2 + 7 – (2x2 – 2) = 9x2 + 7 – 2x2 + 2 Remove parentheses.
= 7x2 + 9 Combine like terms.
2)  (-8y + 5z) – 4(y – 7z) = -8y + 5z – 4y + 28z Remove parentheses.
= -12y + 33z Combine like terms.
3)   – (3a2 + 4a – 4) + 3(4a2 – 6a + 7) Remove parentheses.
= – 3a2 – 4a + 4 + 12a2 – 18a + 21 Distributive property.
= 9a2 – 22a + 25 Combine like terms.
4)   -5(u2  3u) + 3(2u 4) (5 3u + 4u2) Distributive property.
= -5u2 + 15u + 6u – 12 – 5 + 3u 4u2 Remove parentheses.
= -9u2 + 24u – 17 Combine like terms.

 

Multiplying and Dividing Algebraic Expressions

Multiplying a monomial and a polynomial:

  • Use the distributive property: a (b + c) = ab + ac
  • Multiply coefficients and add exponents with the same base.          Apply am an = am+n

Example 1.1.3

1) 3x3 (5x2 – 2x) = (3x3) (5x2) (3x3) (2x) Distributive property:  a (b + c) = ab + ac
= (3 ∙ 5) (x3 x2) (3 ∙ 2) (x3 x1) Regroup           x = x1
= 15 (x3+2) 6 (x3+1) Multiply the coefficients & add the exponents.
= 15x56x4 aman = am+n
2) 5ab2 (2a2b +ab2a) Distribute.
= (5ab2) (2a2b) + (5ab2) (ab2) + (5ab2) (-a) Multiply the coefficients and add exponents.
= (5 ∙ 2) (a1+2 b2+1) + (5a1+1 b2+2) (5a1+1b2) b = b1     ,     a = a1
= 10a3b3 + 5a2b4 – 5a2b2 aman = am+n

 

Dividing a polynomial by a monomial: 

  • Split the polynomial into several parts.
  • Divide a monomial by a monomial.                      Apply \frac{a^m}{a^n}=a^{m-n}.

Example 1.1.4

\frac{12x^2+4x-2}{4x}

Steps                                                               Solution

  • Split the polynomial into three parts:          \frac{12x^2+4x-2}{4x}=\frac{12x^2}{4x}+\frac{4x}{4x}-\frac{2}{4x}
  • Divide a monomial by a monomial:                  =3x+1-\frac{1}{2x}             \frac{a^m}{a^n}=a^{m-n}

 

The FOIL method: an easy way to find the product of two binomials (two terms).

(a + b) (c + d) = ac + ad + bc + bd
                           F      O      I      L
Example
F – First terms first term × first term               (a + b) (c + d) (x + 5) (x + 4)
O – Outer terms outside term × outside term     (a + b) (c + d) (x + 5) (x + 4)
I – Inner terms inside term × inside term         (a + b) (c + d) (x + 5) (x + 4)
L – Last terms last term × last term                 (a + b) (c + d) (x + 5) (x + 4)

 

FOIL method Example
(a + b) (c + d) = ac + ad + bc + bd (x + 5) (x + 4) = x ∙ x + x ∙ 4 + 5x + 5 ∙ 4 = x2 + 9x + 20
                           F     O      I       L                              F        O       I       L

 

Multiplying binomials (2 terms × 2 terms): 

Example 1.1.5

1) (2x + 3)(5x - 6)=2x \cdot 5x+2x(- 6)+3 \cdot 5x+3(-6)

                                         F             O           I             L 

The FOIL method.
=10x^2-12x+15x-18 an am = an+ m
=10x^2+2x-18 Combine like terms.

 

2) (3r - t)(5r+t^2)=3r \cdot 5r+3r \cdot t^2-t \cdot 5r-t \cdot t^2 FOIL
=15r^2+3rt^2-5rt-t^3 an am = an+m
=18r^2+-5rt-t^3 Combine like terms.

 

3) (xy^2+y)(2x^2y+x)=xy^2 \cdot 2x^2y+xy^2 \cdot x+y \cdot 2x^2y+yx FOIL
=2x^3y^3+x^2y^2+2x^2y^2+xy an am = an+m
=2x^3y^3+3x^2y^2+xy  Combine like terms.

 

4) (a-\frac{1}{3})(a-\frac{1}{3})=a^2-\frac{1}{3}a-\frac{1}{3}a+(-\frac{1}{3})(-\frac{1}{3}) FOIL
=a^2-\frac{2}{3}a+\frac{1}{9} Combine like terms.

 

Practice questions                                               

1. Identify the terms of each polynomial:

a. 5x3 + 8x2 + 2x                                                                                             

b. \frac{2}{3}y4 + 9a2 + a – 1                                                                                      

2. Combine like terms:

a. 7x + 10y – 8x + 9y

b. 12a2  33b + 2b – 6a2

c. 13n + 5(6nm2) + 7(2m2 + 3n)

3. Simplify:

a. 15a2 + 9 – (5a2 – 4)

b. (-13x + 9y) – 6(x – 5y)                               

c. 5(ab – 2xy) – 6(-2ab + 3xy)

d. (5y – 7) (8y + 9)

e. (7r – 2t) (3r + 4t2)

f. (x \frac{1}{3}) (x\frac{2}{3})

 

Answers

1. a. 5x3   ,   8x2   ,   2x

    b.\frac{2}{3}y4   ,   9a2   ,   a   ,   –1

2. a.x + 19y

    b. 6a2 – 31b

    c. 9m2 + 64n

3. a. 10a2 + 13

    b. -19x + 39y

    c. 17ab – 28xy

    d. 40y2 11y 63

    e. 21r2 + 28rt2 6rt –8t3

    f. x2 x + \frac{2}{9}