Algebraic rational expressions


Let a be a real number and n be a natural number. The product a · a · a · … · a = aⁿ
is called the nth degree of the number a.

The symbol aⁿ is called a degree, the number a is the basis of the degree, the n-degree indicator or exponent. The degree of the number a with exponent 1 is the number a itself, ie.
a¹ = a


The degree indicator 0 is equal to 1, ie. a⁰ = 1
Degree with an index integer is:


Expressions formed by constants and variables, using the operations of addition, subtraction, multiplication and division of real numbers, as well as scaling with a natural number, are called rational algebraic expressions. Expressions formed as a product of a constant and degrees of a variable are called monomials. A monomial is given in normal form if it contains only one numeric multiplier and does not contain degrees with the same basis.

– For the monomial 2x, the number 2 is called the coefficient of the monomial, and the other part is the principal value.
– Monomials that have the same principal values are called similar.
Degree of a monomial is the sum of the indicators of the degrees of the variables in that monomial.
– The sum of a finite number of monomials is called a whole rational expression or polynomial.
Expressions in which division by a variable occurs or by an expression containing variables are called fractional rational expressions.
– The largest of the degrees of the monomials that make up the polynomial is called the degree of the polynomial.

4.1. Binomial square

(А + В) ² = А² + 2АВ + В²
(А – В) ² = А² – 2АВ + В²

4.2. Product of the sum and difference of two expressions

(А + В) · (А – В) = А² – В²

4.3. Third degree binomial

(А + В) ³ = А³ + ЗА²В + 3АВ² + В³
(А – В) ³ = А³ – 3А²В + 3АВ² – В³

4.4. Dividing a polynomial by a monomial

A polynomial is divisible by a monomial in such a way that each member of the polynomial is divisible by a monomial.

(12х²у³ + 8x⁴y⁶): 4xy² = 3xy + 2x³y⁴

4.5. Dividing a polynomial by a polynomial

The procedure for dividing two polynomials consists of the following:

– the divisor and the divisor are sorted by the exponents of one of the variables that enter them
(from largest to smallest);
– the first term of the divisor is divided by the first term of the divisor and the first term of the quotient is obtained;
– the first term of the quotient is multiplied by the divisor and the obtained product is subtracted from the divisor;
– the first term of the first remainder is divided by the first term of the divisor and the second term of the quotient is obtained. The procedure continues until we get a remainder zero or a remainder whose degree by that variable is less than the degree of the divisor by the same variable.

(5х³ – 8х² + х – 3): (х² – 2) = 5х – 8
± 5х³ + 10х
-8х² + 11х – 3
+ 8×2 ± 16
11x – 19

The quotient of the division is the polynomial 5x – 8, and 11x – 19 is the remainder.


5.1. Decomposition by extracting a common multiplier before parentheses

AH + VH = X (A + B)

5.2. Decomposition using abbreviated multiplication formulas

1. А² – В² = (А – В) · (А + В)

2. А² ± 2АВ + В² = (А ± В) ²

3. А³ ± В³ = (А ± В) · (А² – + АВ + В²)

5.3. Decomposition by grouping members

We will get acquainted with this way of grouping through the following example:
х² – 3х – y² + Зу =
= (х² – y²) – (Зх – Зу) =
= (х – у) · (х + у) – З (х – у) =
= (x – u) · (x + u – 3)


The common divisor of two or more polynomials containing all their common divisors is called the greatest common divisor (NPD) of those polynomials.

NZD of two or more polynomials is found as follows:

– we decompose the given polynomials into simple factors;
– all simple factors that are common to the given polynomials are multiplied by each other and each of them is taken with the smallest indicator that appears in the decomposed polynomials.

Example 2. Find the NZD of the polynomials:
2a² – 2ab and 8a² -8b²

2a² – 2ab = 2a · (a – b);
8a² – 8b² = 8 (a² – b²) = 2³ (a – b) (a + b)

means NZD of 2a² – 2ab and 8a² – 8b² = 2 (a – b)

Polynomials that have no common divisor other than number 1 are called mutually simple polynomials.


A denominator that is a divisor of all other denominators is called the least common denominator (NCC) of a given polynomial.
The NHS of two or more polynomials is found as follows:
– the given polynomials are decomposed into simple factors;
– then a product is formed from all simple factors and each of them is taken with the largest indicator with which it appears in the decomposed polynomials.

Example 2. Find the NHS for polynomials:
x² – y²; x³ – y³; x² – 2xy + y²;
x² – y² = (x – y) (x + y)
x³ – y³ = (x – y) (x² + xy + y²)
x² – 2xy + y² = (x – y) ²
means NHS for all expressions is: (x + y) (x – y) ² (x² + xy + y²)

ALGEBRA Fractions

The expression A / B, where A, B ≠ 0 are polynomials, is called an algebraic fraction. A is called the numerator, and C is the denominator of the fraction.

8.1. Expansion of an algebraic fraction

8.2. Abbreviation of an algebraic fraction


9.1. Addition and subtraction

9.2. Multiplication

9.3. Division

9.4. Double algebraic fractions

– Algebraic rational expressions –

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