**What is algebraic addition?**

The **algebraic addition** It consists of joining several quantities, which can have different signs, in a single resulting quantity, called addition or simply, sum.

Each addend is called *term*so an algebraic sum consists of two or more terms, which can be grouped with parentheses, brackets and braces, the well-known *grouping symbols*.

This addition can be carried out with real numbers, with algebraic expressions, or with a combination of both. Vectors can also be added.

For example, the following is an algebraic addition with integers and grouping symbols:

2 + [– 10 + (−4 + 11 − 17)]

And this other one involves algebraic expressions and real numbers:

4×2 – 4xy + (2/5) x2 – 12xy + 16

The solution of these sums is shown in detail later on (worked examples 6 and 14), but first it is worth reviewing the techniques and properties applicable to their resolution.

**How to solve algebraic sums?**

The first thing that must be taken into account to carry out the algebraic addition is the law or rule of signs:

If you want to add quantities with the same sign, the absolute values are added and the result bears the sign of the quantities.

When adding quantities with different signs, the absolute values are subtracted and the result is given the sign of the quantity with the greater absolute value.

When multiplying or dividing two numbers with the same sign, the result is always positive.

And if you want to multiply or divide two numbers with different signs, the result is negative.

As a reminder, the absolute value of any quantity x, whether numerical or algebraic, is denoted by │x│ and is calculated as follows:

│x│= x, if x > 0

│x│ = −x, if x < 0

For example:

│3│ = 3

│−5│= − (−5) = 5

**Operations hierarchy**

It may be that in an algebraic addition the aforementioned grouping symbols appear, or it is a more complex operation in which, in addition to the addition, a multiplication, division, exponent or root appear.

Then, before carrying out the addition, it is necessary to resort to the hierarchy of operations, to know the order that must be carried out during the resolution:

1.- Eliminate the grouping signs first, starting with the most internal ones.

2.- Solve exponents or roots, if any.

3.- Carry out the multiplications or divisions, in case the operation includes some, always according to the rule of signs stated above.

4.- Once this is done, the algebraic sums are solved, following the guidelines given by the rule of signs.

In the event that there are several operations of the same hierarchy, the solution begins from left to right.

**Important:** any parentheses preceded by the + sign, whether written explicitly or not, can be removed without affecting the sign of the content. But if the parenthesis is preceded by a – sign, then the signs of the content change.

For example:

(–5 + 8 – 13) = – 5 + 8 –13

–(4 + 25 – 76 –1) = – 4 – 25 + 76 +1

**Properties of algebraic addition**

1.- Commutative property: the order of the addends does not alter the sum. That is: a + b = b + a.

2.- Associative property: if the operation consists of more than two terms, the first two can be associated, obtain their result, add it to the next one, and so on. Therefore:

(a + b) + c = a + (b + c)

3.- Neutral element of the addition: it is 0, therefore: a + 0 = a

4.- Opposite: given the quantity «a», its opposite is «-a», to fulfill that: a + (-a) = 0

5.- When there is a mixed expression, which consists of numbers and algebraic terms, only those that are similar are added and the sum of the non-similar terms is indicated.

Like terms are those whose literal part is identical, although they may differ in the coefficient. For example:

1 + x2 – 4×2 – 7 = (1–7) + (x2 – 4×2) = – 6 – 3×2

The terms x2 and 4×2 are similar, since they have the same letter and exponent. Also note that numbers are added apart from literal expressions (with letter) and the result is left indicated.

**examples**

**Algebraic addition of integers**

There are several strategies, applying the rules of signs and properties described above. For example, you can add the positive and negative quantities separately, and then subtract the respective results.

**1)** 7− 8 + 4 − 10 − 25 + 4 = (7 + 4 + 4) + (− 8 −10 − 25) = 15 + (−43) = − 28

**2)** −15 + 7 − 13 − 34 + 18 −24−26 = (7 + 18) + (−15 − 13 − 34 − 24 − 26) = 25 + (−112) = − 87

**3)** [83 + (–99)] + 18 = –16 + 18 = 2

**4) **21 – 3 – 7 + 20 + 9 – 10 + 15 – 25 + 10 = (21 + 20 + 9 + 15 + 10) + (– 3 – 7– 10 – 25) = 75 – 45 = 30

In the following exercise, keep in mind that a grouping sign preceded by a minus sign changes the sign of the content:

**5)** 9 – [3 – (–9 + 8 + 21)] – 27 = 9 – [3 + 9 – 8 –21] – 27 = 9 – 3 – 9 + 8 + 21 – 27 = (9 + 8 + 21) + (– 3 – 9 – 27) = 38 – 39 = – 1

**6)** 2 + [ – 10 + (−4 + 11 − 17)] = 2 + [ – 10 − 4 + 11 − 17] = 2 + [11+ (– 10 − 4 − 17)] = 2 + [11+ (– 31)] = 2 + (– 20)= – 18

**7)** The Roman Emperor Augustus began his reign in –27 BC and ruled until his death, for 41 years. The year in which Augustus’ reign ended was:

− 27 + 41 = 14 AD

**8)** The elevator of a building is located in the second basement, it goes up seven floors, it goes down four, it goes up 15 and it goes down 6. On which floor is the elevator located?

First the signs are assigned: level 0 at street level, when the elevator goes up a certain number of floors it is considered a positive quantity and when it goes down it is negative:

−2 + 7 − 4 + 15 − 6 = (7 + 15) + (−2− 4− 6) = 22 – 12 = +10

The elevator is on the tenth floor.

**Algebraic addition of real numbers**

The real numbers include the natural, rational, and irrational numbers:

**9) **4 − 3⅚ − √2 + 6√2 + ½ + 11 = (4 + 11) + ( ½ − 3⅚) + (6√2− √2) = 15 + (–10/3) + 5√2 = 35 /3 + 5√2

**10) **3 − 5.5 + (−8.7) =3 − 5.5 − 8.7 = −11.2

**Addition of monomials and polynomials**

The monomials contain a literal part with its respective exponent, which is an integer greater than 1, and a numerical coefficient belonging to the set of real numbers. The literal part can consist of one or more letters.

The expressions: −3×2, √5∙ x3 and 8x2y3 are examples of monomials. Instead, they are not monomials: 2x−3 and 7√x.

Algebraic additions between monomials can only be executed when the monomials are similar, in which case, the result is another monomial. This procedure is also called *reduction of monomials*:

**eleven) **(3/2)∙x3y + 2∙x3y =(7/2)∙x3y

If the monomials are not similar, the sum is indicated and results in a polynomial:

**12) **1 + 6x − 5×2 = 1 + 6x − 5×2

**13)** (√3 x8 + 4x) + (5×8 + 3x) = (√3 x8 + 5×8 ) + (4x + 3x) = (√3 + 5)⋅x8 + 7x

If like terms appear in a sum, they can be reduced:

**14) **4×2 – 4xy + (2/5) x2 – 12xy + 16 = (4×2 + (2/5) x2 )+ (– 4xy – 12xy)+ 16 =(22/5)x2 – 16xy + 16

**fifteen) **3×2 + 5x − 2×2 − 9x = (3×2 − 2×2)+ (5x − 9x) = x2 − 4x

**16) **5×3 –7x + 2x – 9×2 + 2×3 – 5×2 = (5×3 +2×3) + (– 9×2 – 5×2 ) + (–7x + 2x) = 7×3– 14×2 – 5x

The addition of polynomials can be carried out horizontally, as in the previous examples, or vertically. The result is the same in both cases.

**17) **Add polynomials in two ways:

5x² + 7y − 6z²

4y + 3x²

9x² + 2z² − 9y

2y − 2x²

**horizontally**:

(5x² + 7y − 6z²) + (4y + 3x²) + (9x² + 2z² − 9y) + (2y − 2x²) = (5x² + 3x² + 9x² − 2x²) + (− 6z² +2z²) + (7y + 4y − 9y + 2y) = 15x²− 4z² + 4y

**upright**:

+ 5x² + 7y − 6z²

+ 3x² + 4y

+ 9x² − 9y + 2z²

−2x² + 2y

_______________________

+ 15x² + 4y − 4z²

**18) **(1/2 x2 + 4 ) + (3/2 x2 + 5 ) + (x2 + 2) = (1/2 x2 + 3/2 x2 + x2) + (4 + 5 + 2) =

**19) **(3×2 − 5x +1) + (x2 −7x−3) = (3×2 + x2) + (− 5x −7x) + (1 − 3) = 4×2 −12x − 2

**twenty) **Perform the sum of the polynomials:

P(x) = 3×4 + 3×2 − 5x + 7

Q(x)= 2×5 − x4 + x3 − 2×2 + x – 3

R(x) = − 3×5 + 2×4 + 2×3 − 4x − 5

Using the vertical method, the polynomials are completed with the help of terms of the form 0xn and we proceed to add like terms:

0x5 + 3×4 + 0x3 + 3×2 − 5x + 7

2×5 − x4 + x3 − 2×2 + x − 3

−3×5 +2×4 + 2×3 + 0x2 − 4x − 5

_______________________________

−x5 + 4×4 + 3×3 + x2 − 8x − 1