This document provides an overview of algebraic expressions and operations with polynomials. It defines algebraic expressions as combinations of letters and numbers using operations like addition, subtraction, multiplication, and exponents. Monomials are single-term expressions while polynomials can have multiple terms. The key operations discussed are adding and multiplying polynomials by distributing terms. Factoring polynomials uses techniques like removing common factors, applying identities like the difference of squares, and recognizing perfect square trinomials.

1. BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA
ALGEBRA
1. ALGEBRAIC EXPRESSIONS
Algebra is the handling of numerical relations in which one or more quantities are unknown.
These terms are called variables or unknowns and they are represented by letters.
An algebraic expression is a combination of letters and numbers linked by the signs of
operations: addition, subtraction, multiplication and powers.
EXAMPLES: The algebraic expressions allow us, for example to find areas and volumes.
Length of the circumference: L = 2 r, where r is the radius of the circumference.
Area of the square: S = l2
, where l is the side of the square.
EXAMPLES OF COMMON ALGEBRAIC EXPRESSIONS:
The double of a number: 2x. The triple a number: 3x Half of a number: x/2.
A third of a number: x/3. A quarter of a number: x/4. A number to the square: x2
Two consecutive numbers: x and x + 1.
2. NUMERICAL VALUE OF AN ALGEBRAIC EXPRESSION
The numerical value of an algebraic expression for a particular value is the number
obtained by replacing the unknown values with the numerical value given and perform the
operations.
EXAMPLE: f(x)= x3
x = 5 cm f(5) = 53
= 125
3. TYPES OF ALGEBRAIC EXPRESSIONS
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2. A monomial is an algebraic expression formed by a single term, in which the only
operations that appear between the variables are the product and the power of a natural
exponent. Example: 2x2
y3
z.
A binomial is an algebraic expression formed by two terms.
A polynomial is an algebraic expression consists of more than one term.
4. PARTS OF A MONOMIAL
The coefficient of a monomial is the number that multiplies the variable(s).
The literal part is constituted by the letters and its exponents.
The degree of a monomial is the sum of all exponents of the letters or variables.
gr(2x2
y3
z) = 2 + 3 + 1 = 6
Similar monomials: Two monomials are similar when they have the same literal part.
2x2
y3
z is similar to 5x2
y3
z
5. OPERATIONS WITH MONOMIALS
5.1 Addition of Monomials:
We can add monomials if they are similar. The sum of the monomials is another monomial that
has the same literal part and whose coefficient is the sum of the coefficients.
EXAMPLE: 2x2
y3
z + 3x2
y3
z = 5x2
y3
z
If the monomials are not similar a polynomial is obtained: x2
y3
+ 3x2
y3
z
5.2 Multiplication of a Number by a Monomial
The product of a number by a monomial is another similar monomial whose coefficient
is the product of the coefficient of the monomial and the number.
EXAMPLE: 5 · (2x2
y3
z) = 10x2
y3
z
5.3 Multiplication of Monomials
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3. The multiplication of monomials is another monomial that takes as its coefficient the product
of the coefficients and whose literal part is obtained by multiplying the powers that have the
same base.
EXAMPLE: (5x2
y3
z) · (2 y2
z2
) = 10 x2
y5
z3
5.4 Division of Monomials
Dividing monomials can only be performed if they have the same literal part and the degree
of the dividend has to be greater than or equal to the corresponding divisor.
The division of monomials is another monomial whose coefficient is the quotient of the
coefficients and its literal part is obtained by dividing the powers that have the same base.
EXAMPLE:
If the degree of divisor is greater, an algebraic fraction is obtained.
EXAMPLE:
5.5 Power of a Monomial
To determine the power of a monomial, every element in the monomial is raised to the
exponent of the power.
EXAMPLE: (2x3
)3
= 23
· (x3
)3
= 8x9
(−3x2
)3
= (−3)3
· (x2
)3
= −27x6
6. POLYNOMIALS
A polynomial is an algebraic expression in the form:
P(x) = an xn
+ an - 1 xn - 1
+ an - 2 xn - 2
+ ... + a1 x1
+ a0
an, an -1 ... a1 , ao... are the numbers and are called coefficients
n is a natural number.
x is the variable.
ao is the independent term.
The degree of a polynomial P(x) is the greatest degree of the monomials.
Two polynomials are similar if they have the same literal part.
EXAMPLE: P(x) = 2x3
+ 5x − 3 and Q(x) = 5x3
− 2x − 7
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4. 7.OPERATIONS WITH POLYNOMIALS
6.1 Adding Polynomials
To add two polynomials, add the coefficients of the terms of the same degree.
EXAMPLE: P(x) = 2x3
+ 5x − 3 Q(x) = 2x3
− 3x2
+ 4x
P(x) + Q(x) = (2x3
+ 5x − 3) + (2x3
− 3x2
+ 4x) = 2x3
+ 2x3
− 3x2
+ 5x + 4x − 3= 4x3
− 3x2
+ 9x − 3
Polynomials can also be added by writing them under each other, so that similar monomials
that are in the same columns can be added together.
P(x) = 7x4
+ 4x2
+ 7x + 2 Q(x) = 6x3
+ 8x +3
P(x) + Q(x) = 7x4
+ 6x3
+ 4x2
+ 15x + 5
6.2 Multiplication of a Number by a Polynomial
It is another polynomial that has the same degree. The coefficients are the product of the
coefficients of the polynomial and the number.
EXAMPLE: 3 · (2x3
− 3x2
+ 4x − 2) = 6x3
− 9x2
+ 12x − 6
6.3 Multiplying a monomial by a Polynomial
The monomial is multiplied by each and every one of the monomials that form the
polynomial.
EXAMPLE: 3x2
· (2x3
− 3x2
+ 4x − 2) = 6x5
− 9x4
+ 12x3
− 6x2
6.4 Multiplication of Polynomials
Multiply each monomial from the first polynomial by each of the monomials in the second
polynomial. The multiplication of polynomials is another polynomial whose degree is the
sum of the degrees of the polynomials that are to be multiplied.
EXAMPLE: P(x) = 2x2
− 3 Q(x) = 2x3
− 3x2
+ 4x
P(x) · Q(x) = (2x2
− 3) · (2x3
− 3x2
+ 4x) =
= 4x5
− 6x4
+ 8x3
− 6x3
+ 9x2
− 12x = 4x5
− 6x4
+ 2x3
+ 9x2
− 12x
The polynomials can also be multiplied as follows:
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5. 8. ALGEBRAIC IDENTITIES
Square of a Binomial: (a ± b)2
= a2
± 2 · a · b + b2
EXAMPLES : (x + 3)2
= x 2
+ 2 · x ·3 + 32
= x 2
+ 6 x + 9
(2x − 3)2
= (2x)2
− 2 · 2x · 3 + 32
= 4x2
− 12 x + 9
Difference of Squares: (a + b) · (a − b) = a2
− b2
EXAMPLE: (2x + 5) · (2x - 5) = (2x)2
− 52
= 4x2
− 25
Cube of a Binomial: (a ± b)3
= a3
± 3 · a2
· b + 3 · a · b2
± b3
EXAMPLES: (x + 3)3
= x3
+ 3 · x2
· 3 + 3 · x · 32
+ 33
= x 3
+ 9x2
+ 27x + 27
(2x − 3)3
= (2x)3
− 3 · (2x)2
·3 + 3 · 2x · 32
− 33
= 8x 3
− 36x2
+ 54x − 27
Square of a Trinomial: (a + b + c)2
= a2
+ b2
+ c2
+ 2 · a · b + + 2 · a · c + 2 · b · c
EXAMPLE: (x2
− x + 1)2
= (x2
)2
+ (−x)2
+ 12
+ 2 · x2
· (−x) + 2 x2
· 1 + 2 · (−x) · 1= x4
+ x2
+ 1 − 2x3
+ 2x2
− 2x= x4
− 2x3
+ 3x2
− 2x + 1
9. METHODS FOR FACTORING A POLYNOMIAL
9.1.Remove the Common Factor: It consists by applying the distributive property.
a · b + a · c + a · d = a (b + c + d)
EXAMPLES. x3
+ x2
= x2
(x + 1)
2x4
+ 4x2
= 2x2
(x2
+ 2)
x2
− ax − bx + ab = x (x − a) − b (x − a) = (x − a) · (x − b)
9.2 Remarkable Identities:
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6. A difference of squares is equal to the sum of the difference. a2
− b2
= (a + b) · (a − b)
EXAMPLES:
x2
− 4 = (x + 2) · (x − 2)
x4
− 16 = (x2
+ 4) · (x2
− 4) = (x + 2) · (x − 2) · (x2
+ 4)
A perfect square trinomial is equal to a squared binomial. a2
± 2 a b + b2
= (a ± b)2
.
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7. A difference of squares is equal to the sum of the difference. a2
− b2
= (a + b) · (a − b)
EXAMPLES:
x2
− 4 = (x + 2) · (x − 2)
x4
− 16 = (x2
+ 4) · (x2
− 4) = (x + 2) · (x − 2) · (x2
+ 4)
A perfect square trinomial is equal to a squared binomial. a2
± 2 a b + b2
= (a ± b)2
.
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