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CONTENTS

ALGEBRAIC EXPRESSIONS

TERM (MONOMIAL)

A term is a numerical

constant or the product

(or quotient) of a

numerical constant and

one or more variables.

POLYNOMIALS

A polynomial is a

monomial or a sum of

monomials.

BINOMIALS AND TRINOMIALS

A binomial is a sum of two

monomials, and a

trinomial is a sum of three

monomials.

EVALUATING AN EXPRESSION

To evaluate an algebraic

expression, plug in the

unknowns and calculate

according to PEMDAS.

Find the value of when x = −2.

Plug in −2 for x:

2 5 6x x

2

2 5 2 6

4 10 6

12

LIKE TERMS

Two algebraic expressions

are called like terms if

both the variable parts

and the exponents are

identical.

ALGEBRAIC EXPRESSIONS

OPERATIONS WITHALGEBRAIC EXPRESSIONS

ADDING/SUBTRACTING MONOMIALS

To combine like terms,

keep the variable part

unchanged while

adding or subtracting

the coefficients.

ADDING/SUBTRACTING POLYNOMIALS

To add or subtract polynomials,

combine like terms.

5 2 1 2 7 4

5 2 1 2 7 4

3 5 3

x y x y

x y x y

x y

MULTIPLYING MONOMIALS

To multiply monomials,

multiply the

coefficients and the

variables separately.

MULTIPLYING BINOMIALS – FOIL

To multiply binomials,

use FOIL, then add

and combine like

terms.

MULTIPLYING POLYNOMIALS

Multiply each

term in the first

polynomial by

each term in the

second.

(3g – 3) (2g2 + 4g – 4)

6g3 + 12g2 – 12g – 6g2 – 12g + 12

6g3 + 6g2 – 24g + 12

2 2 2

2 2 2

2 2

2

2

a b c ab ac

a b c ab ac

a b a ab b

a b a ab b

a b a b a b

SPECIAL PRODUCTS

22 2

22 2

2 2

2

2

2

5 6 2 3

ab ac a b c

ab ac a b c

a ab b a b

a ab b a b

a b a b a b

a a a a

FACTORING

mm m

mm m

a b a b

a b a b

MULTIPLYING RATIONAL EXPRESSIONS

To multiply fractions,

you multiply the

numerators and

multiply the

denominators.

DIVIDING RATIONAL EXPRESSIONS

To find the quotient

of two fractions,

you multiply by the

reciprocal of the

divisor.

OPERATIONS WITH

ALGEBRAIC EXPRESSIONS

SOLVING EQUATIONS

SOLVING EQUATIONS

The solutions of an equation or

inequality with one or more

unknowns are those values

that make the equation true.

The solutions of an equation

“satisfy the equation or

inequality”, when they are

substituted for the unknowns of

the equation or inequality.

Solutions Roots

An equation may have no

solution or one or more

solutions.

If a = b, then:

a c b c

a c b c

a c b c

, 0a b

cc c

2 2a b

, , 0a b a b

1 1, , 0a b

a b

SOLVING EQUATIONS “IN TERMS OF …”

Isolate one variable on

one side of the equation,

leaving an expression

containing the other

variable in the other side

of the equation.

10y x

SOLVINGEQUATIONS

SOLVING SYETMS OF EQUATIONS

SOLVING SYSTEMS OF EQUATIONS

If two or more equations are

to be solved together, the

solutions must satisfy all the

equations simultaneously.

No solution Multiple solutionsOne solution

You can solve for

two variables only

if you have two

distinct equations.

Two forms of the

same equation will

not be adequate.

5 15

5 15

y x

y x

Combine the equations in

such a way that one of

the variables cancels out,

isolate the variable, and

then plug that expression

into the other equation.

3

4

5

a b

a c

b c

3

4

5

a b

a c

b c

Find the value of a + b + c.

2 2 2 12a b c

6a b c

SUSTITUTION

Solve for x and y: 2 13, 2x y x y

2x y 2( 2) 13

2 4 13

3 13 4

3 9

3

y y

y y

y

y

y

2 3 13

2 13 3

2 10

5

x

x

x

x

ELIMINATION

Solve for x and y: 2 8, 1x y x y

2 8

1

3 9

3

x y

x y

x

x

2 3 8

6 8

8 6

2

y

y

y

y

SOLVING A QUADRATIC FACTORABLE EQUATION

To solve a quadratic equation, put

it in the “ ax2 + bx + c = 0 ” form.

SOLVING A QUADRATIC FACTORABLE EQUATION

Factor the left side (if you can),

and set each factor equal to 0

separately to get the two solutions.

The solutions of an equation are

also called roots.

Every positive number

has two square roots.

If x2 = 16, then x = 4 or x = 4.

1 and 2 are the roots of the equation

x2 + 3x + 2 = 0, because:

(1)2 + 3(1) + 2 = 0, and,

(2)2 + 3(2) + 2 = 0.

QUADRATIC EQUATION FORMULA

For , where ,

the value of x is given by:

2 0ax bx c 0a

2 4

2

b b acx

a

Solve for x: 2 11 30 0x x

2

1 2

11 11 4 1 30

2 1

11 1

2

6, 5

x

x

x x

2

1, 11, 30

4

2

a b c

b b acx

a

PROPERTIES OF THE ROOTS

If the roots of a quadratic equations

are r1 and r2, the equation can be

written as

1 2 0x r x r

In a quadratic equation of the form

, the sum of the

roots is equal to –d and the

product of the roots is equal to e.

2 0x dx e

1 2

1 2

r r d

r r e

Solve for x: 2 11 30 0x x

2

1

2

11 30 0

5 6 0

5 0 5

6 0 6

x x

x x

x x

x x

1 2

1 2

11,

30

x x

x x

Note that:

To solve an exponential equation,

put both sides of the equation in the

form ax, and make a new equation

using just the exponents.

SOLVING EXPONENTIAL

EQUATIONS

If ax = ay, then x = y ( a ≠ 0 and a ≠ 1).

32 2

3

x

x

SOLVING EQUATIONS

SOLVING INEQUALITIES

INEQUALITIES

4 7

4 < x < 7

4 ≤ x ≤ 7

4 7

4 < x ≤ 7

4 7

4 ≤ x < 7

4 7

4 ≤ x

4

4 < x

4

x ≤ 7

7

x < 7

7

SOLVING INEQUALITIES

If a > b, then:

a c b c

a c b c

0,If c a c b c 0,

a bIf c

c c

1 1, , 0a b

a b

0,If c a c b c 0,a b

If cc c

Isolate the variable. 5 7 3

5 3 7

5 10

10

5

2

x

x

x

x

x

6 52 5

3

3 2 5 6 5

xx

x x

6 15 6 5

15 6 6 5

9 1

1

9

x x

x x

x

x

SOLVING INEQUALITIES

SUMMARY

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