We will explain algebraic operations and how to do them because of the great importance of carrying out such operations. It is recommended that the reader practice these operations to increase his ability in algebra.
When writing all the operations that we will see next, I will tell you that we use grouping symbols, the best known is the parenthesis ( \ ). Which means that the terms enclosed in parentheses must be considered as a single number. I’ll show you more grouping symbols: square brackets [ \ ], braces \{ \ \} and the bar \overline{\ \ \ \ \ \ \ } which is placed above the terms to be grouped: 1 + \overline{2 + 3}.
Algebraic expression, term and polynomial
An algebraic expression is so named because it is the result of a process. Just as 2x^{3} - y is an algebraic expression because it is formed by performing algebraic operations with the numbers 2 and 3 and the letters x and y. Examples of algebraic expressions are the following:
\cfrac{4y}{7x} + 2z^{2}, \qquad \quad \cfrac{x + 3}{x^{4} + 3x^{2} + 1}
The simplest of expressions is obtained by combining letters and numbers from any of the algebraic operations excluding addition and subtraction. Each of them is called an algebraic term, let’s see several examples:
\cfrac{2x}{3y}, \ 10xy, \ 8\sqrt{ab}, \ 8x, \ xy
Every algebraic term has a factor called the coefficient, the term 8\sqrt{ab} has a coefficient of 8, the term \frac{2x}{3y} has a coefficient of \frac{2}{3}.
Algebraic terms that are only different in their coefficients are called similar terms. The following terms are similar:
ab \quad \text{y} \quad -6ab
The degree of a term is understood to be the sum of the aforementioned exponents. For example, the term xyz is of third degree, the term \sqrt{2}a^{3}bc^{2} is of 6th degree. Look at some more examples:
\begin{array}{c c c} \ \text{1st degree} \ & \ \text{4th degree} \ & \ \text{7th degree} \ \\ y & xy^{3} & nx^{2}y^{3}z \end{array}When you have only one term, it is called a monomial. If there are two or more terms linked by the signs of addition (+) and subtraction (-), it is called algebraic addition or subtraction, at the same time, if we have two terms that are added or subtracted, that expression is known as a binomial, if we have three terms it is known as a trinomial, and from here on, the more terms there are, the expression will be known as a multinomial or more preferably a polynomial.
\begin{array}{c c c c} \ \text{Monomial} & \text{Binomial} & \text{Trinomial} & \text{Polynomial} \ \\ y & x + y & \ x + y + z \ & \ n + x + y + z \end{array}If all the terms of a polynomial are of the same degree, it is known as a homogeneous expression.
\begin{array}{c c c} \ \text{Homogeneous} \ & \ \text{Homogeneous} \ & \ \text{Not homogeneous} \ \\ x^{2} + y^{2} & xyz + n^{3} & xy + z \end{array}Addition
There are 5 properties or laws of addition.
Law of existence
Addition is always possible. It is always possible to add two or more numbers, resulting in another number.
Law of uniqueness
Given two numbers a and b, there is a single number c such that a + b = c.
Zero is a number, but this law does not apply when we want to add any number a with zero. Since a will remain a: a = a. This tells us that it is an important property of zero.
Commutative law
If a and b are two numbers, then a + b = b + a. The sum of two or more numbers is independent of the order in which they are written:
1 + 2 = 2 + 1
Associative law
If a, b, and c are three numbers, then \left(a + b \right) + c = a + \left (b + c \right). The sum of three or more numbers is independent of the way they are grouped:
\left( 1 + 2 \right) + 3 = 1 + \left( 2 + 3 \right) = 2 \left( 3 + 1 \right)
Additive law of equality
If a, b, and c are any numbers such that a = b, then a + c = b + c.
Subtraction
Subtraction is the inverse operation of addition. Having the following property:
Subtractive property of equality
If a, b, and c are any numbers such that a = b, then a - c = b - c.
Next we will talk about a subtraction theorem.
Subtraction theorem 1
The sum of any negative number with its corresponding positive number equals zero.
(-a) + a = (0 – a) + a = 0 – a + a = 0
There are more theorems, but we only consider placing the one seen above since it is the most relevant. We greatly appreciate Lehmann’s algebra book, but we will not put all the theorems he mentions in subtraction because we do not think it is very necessary to mention them.
Multiplication
Let’s look at the laws of multiplication
Law of existence of multiplication
Multiplication is always possible. Multiplying any two or more numbers will also result in a number.
Law of uniqueness of multiplication
Multiplication is unique. For any two numbers a and b, there is only one number c such that a \cdot b = c. Also, c is the product of a times b and a and b are the factors of c. And a and b are called multiplicand and multiplier, respectively.
Commutative law of Multiplication
If a and b are any two numbers then a \cdot b = b \cdot a . The order of the factors does not alter the product.
3 \cdot 4 = 4 \cdot 3 = 12
Associative law of Multiplication
If a , b and c are any three numbers, then:
c\cdot (a\cdot b) = a\cdot (b \cdot c) = b \cdot (a\cdot c)
The multiplicative property of equality
If a , b and c are any three numbers such that a = b , then:
a\cdot c = b\cdot c
The distributive property of multiplication
If a , b and c are any three numbers, then:
a\cdot (b + c) = a\cdot b + a \cdot c \ \rightarrow \ 2(2 + 4) = 2 \cdot 2 + 2 \cdot 4
Multiplication theorem 1
The product of any number and zero equals zero.
a\cdot 0 = a \cdot (c - c) = a\cdot c - a\cdot c = 0
Multiplication theorem 2
The product of a positive number and a negative number is a negative number.
a\cdot (-b) = x \ \rightarrow \ 3\cdot (-2) = -6
Multiplication theorem 3
The product of two negative numbers is a positive number.
(-a)(-b) = x \ \rightarrow \ (-3)(-4) = 12
Multiplication theorem 4
If the product of two numbers is equal to zero, at least one of the factors is equal to zero.
a\cdot b = 0
Division
The division can be represented by the following equality:
c = \cfrac{b}{a}
What is read: c is equal to b divided by a, which means that c is the quotient obtained by dividing the dividend b by the divisor a. We can also say that c is the number by which a must be multiplied to obtain the product b. This is how we have the following equality:
a\cdot \cfrac{b}{a} = b \qquad \quad a \neq 0
I will tell you at once that the division can be represented as follows:
\cfrac{b}{a}, \quad b/a, \quad b \div a, \quad b:a
Next we will observe some theorems that will be very useful when doing divisions.
Division theorem 1
Division by the number zero is impossible.
Using a little logic, take a number and divide it by 2, then take the same number and divide it by a smaller number, and so on until you see what happens.
\cfrac{1}{2} = 0.5 \qquad \cfrac{1}{0.5} = 2 \qquad \cfrac{1}{0.01} = 100
If we divide the same number by a number very close to zero, we will obtain a number so large that it would be impossible to write it. Now try to imagine what will happen if we divide by zero.
That is a simple explanation why we cannot divide by zero.
Division theorem 2
If zero is divided by any nonzero number, the quotient (result) is always zero.
b = \cfrac{0}{a} \ \rightarrow \ b = 0
Division theorem 3
The product of two quotients w/x and y/z is another quotient.
\cfrac{a}{c} \cdot \cfrac{b}{d} = \cfrac{a\cdot b}{c \cdot d}
Division theorem 4
If a and b are both different from zero, and m, n, r, and s are positive integers such that m>n and r>s, then :
\cfrac{a^{m}b^{r}}{a^{n}b^{s}} = a^{m-n}b^{r-s}
Let’s see some examples:
\cfrac{a^{4}b^{2}}{a^{3}b^{4}} = \cfrac{a^{4}}{a^{3}} \cdot \cfrac{b^{2}}{b^{4}} = a^{4-3}b^{2-4} = a\cdot b^{-2} = \cfrac{a}{b^{2}}
\cfrac{10 a^{3}b^{2}}{2 a^{2}b^{2}} = \cfrac{10}{2} \cdot \cfrac{a^{3}}{a^{2}} \cdot \cfrac{b^{2}}{b^{2}} = 5 \cdot a^{3-2} \cdot b^{2-2} = 5 \cdot a \cdot 1 = 5a
Division theorem 5
To divide a polynomial by a monomial, each term of the polynomial is divided by the monomial, and the quotients obtained are added.
\cfrac{a + b – c + d}{n} = \cfrac{a}{n} + \cfrac{b}{n} - \cfrac{c}{n} + \cfrac{d}{n}
Exponents
The laws regarding exponents that we must know are the following:
- x^{m}\cdot a^{n} = x^{m+n}
- \left( x^{m} \right)^{n} = a^{m\cdot n}
- \left( x \cdot y\right)^{m} = x^{m} \cdot y^{m}
- \left( \cfrac{x}{y} \right)^{m} = \cfrac{x^{m}}{y^{m}}
- \cfrac{x^{m}}{y^{m}} = x^{m - y}, \qquad m > n
- \cfrac{x^{m}}{x^{n}} = \cfrac{1}{x^{n - m}}, \qquad m < n
Radicals
The expression representing radicals is \sqrt[n]{a}, where q is called the index and a is called the radicand. We can express the radical in two different ways, notice:
a^{\frac{1}{n}} = \sqrt[n]{a}
Laws of radicals
- \sqrt[n]{a} \cdot \sqrt[n]{a} = \sqrt[n]{a\cdot b}
- \cfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\cfrac{a}{b}} \qquad b\neq 0
- \sqrt[m]{\sqrt[n]{a}} = \sqrt[n]{\sqrt[m]{a}} = \sqrt[mn]{a}
Simplify radicals
We use these laws to simplify the radicals, which are said to be simplified when they fulfill the following:
- The subradical does not contain affected factors of exponents greater than the index q of the radical.
- The subradical does not contain fractions.
- The index of the radical is the smallest possible.
Now let’s look at each example of:
Addition and subtraction of radicals
To apply addition and subtraction to radicals, you need two or more radicals to be similar. Example, 4\sqrt{2} and 5\sqrt{2} are like radicals. We will add or subtract as if the root were an unknown. Let’s see an example, simplify the following expression:
4\sqrt{2} + 8\sqrt{8} - 3\sqrt{18}
To simplify the expression, we first need to break down each of the terms:
4\sqrt{2} + 8 \sqrt{2\cdot 2 \cdot 2} - 3 \sqrt{3\cdot 3 \cdot 2}
Now we will represent as squared the terms that can be, as can be seen in the second term, we can express those numbers two as 2^{2} \cdot 2 and the same with the three of the third term:
4\sqrt{2} + 8\sqrt{2^{2}\cdot 2} - 3 \sqrt{3^{2} \cdot 2}
And the terms that have the same number in the exponent as in the index of the radical that contains them, can come out of said radical. As we have a 2 ^ {2} that has the same exponent as the square root that contains it, then we can take it from the square root and it will become simply a 2 , the same applies to all the radicals that we have in the expression :
4\sqrt{2} + 8\cdot 2 \sqrt{2} - 3 \cdot 3 \sqrt{2}
Now we carry out the multiplications:
4\sqrt{2} + 16\sqrt{2} - 9 \sqrt{2}
And we carry out the addition and subtraction as if the radicals were an unknown, which means that we will add 4 + 16 and then subtract the 9 :
Notable products
Click here to see the full Notable Products post.
Factoring
Click here to see the full post of Factorization
Least common multiple
The least common multiple is a polynomial that is divisible into whole parts by another that is called a multiple. For example, x^{2} - y^{2} is a common multiple of x + y and x - y.
That common multiple of two or more polynomials that has the lowest possible degree is called the least common multiple and it is usually designated by the acronym of L. C. M.
The L.C.M. \ge of two polynomials is equal to the product of all the different factors of these polynomials and each factor with the maximum exponents will be taken.
The least common multiple will be the multiplication of all the numbers that we use to divide the numbers they give us, let’s see an example.
Example of least common multiple
Determine the least common multiple of x^{2} - y^{2}, x^{3} - y^{3} and x^{2} + 2xy + y^{2}. Let’s use the famous table of the least common multiple:
\begin{array}{c c c c c | c} x^{2} + 2xy + y^{2} & \quad & x^{3} - y^{3} & \quad & x^{2} - y^{2} & 1 \\ \hline & & & & & \\ & & & & & \\ & & & & & \end{array}We will write a “number” to divide some polynomial that we have, it doesn’t matter if it doesn’t divide all of them:
\begin{array}{c c c c c | c} x^{2} + 2xy + y^{2} & \quad & x^{3} + y^{3} & \quad & x^{2} - y^{2} & \\ \hline x + y & & x^{2} - xy + y^{2} & & x - y & x + y \\ & & & & & \\ & & & & & \end{array}And so we will continue until we manage to divide until all the polynomials become 1.
\begin{array}{c c c c c | c} x^{2} + 2xy + y^{2} & \quad & x^{3} + y^{3} & \quad & x^{2} - y^{2} & \quad \\ \hline x + y & & x^{2} - xy + y^{2} & & x - y & x + y \\ 1 & & x^{2} - xy + y^{2} & & x -y & x + y \\ & & & & & \end{array}\begin{array}{c c c c c | c} x^{2} + 2xy + y^{2} & \quad & x^{3} + y^{3} & \quad & x^{2} - y^{2} & \quad \\ \hline x + y & & x^{2} - xy + y^{2} & & x - y & x + y \\ 1 & & x^{2} - xy + y^{2} & & x -y & x + y \\ 1 & & 1 & & x - y & x^{2} - xy + y^{2} \\ & & & & & \end{array}\begin{array}{c c c c c | c} x^{2} + 2xy + y^{2} & \quad & x^{3} + y^{3} & \quad & x^{2} - y^{2} & \quad \\ \hline x + y & & x^{2} - xy + y^{2} & & x - y & x + y \\ 1 & & x^{2} - xy + y^{2} & & x -y & x + y \\ 1 & & 1 & & x - y & x^{2} - xy + y^{2} \\ 1 & & 1 & & 1 & x - y \end{array}And the result of the least common multiple are all those polynomials that we have to the right of our table.
\text{least common multiple} = \left( x + y \right)^{2}(x - y)\left(x^{2} - xy + y^{2} \right)Let’s see an example of the least common multiple with numbers
Determine the least common multiple of the numbers 4, 20, 37 and 45. We will write the table and then we will write a second table with the results. We recommend that you try to do it and then check if you have the same result as us.
\begin{array}{c c c c c c c | c} 6 & \quad & 20 & \quad & 47 & \quad & 45 & \quad \\ \hline & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \\ & & & & & & & \end{array}The response box is as follows:
\begin{array}{c c c c c c c | c} 6 & \quad & 20 & \quad & 47 & \quad & 45 & \quad \\ \hline
2 & & 20 & & 47 & & 15 & 3 \\ 1 & & 10 & & 47 & & 15 & 2 \\ & & 2 & & 47 & & 3 & 5 \\ & & 2 & & 1 & & 3 & 47 \\ & & 2 & & & & 1 & 3 \\ & & 1 & & & & & 2 \end{array}The least common multiple of the numbers 6 , 20 , 47 and 45 is:
\text{least common multiple } = 3 \cdot 2 \cdot 5 \cdot 47 \cdot 3 \cdot 2 = 8460
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