At algebraic expressions are formed by three basic items: known numbers, unknown numbers and math operations. At numeric expressions and algebraic follow the same order of resolution. In this way, operations inside parentheses have priority over others, as well as multiplications and divisions take precedence over additions and subtractions.
Unknown numbers are called incognitos and are usually represented by letters. Some books and materials also call them variables. The numbers that accompany these incognitos are called coefficients.
Therefore, examples of algebraic expressions are:
1) 4x + 2y
2) 16z
3) 22x + y - 164x2y2
Numerical value of algebraic expressions
when the unknown it is no longer an unknown number, just replace its value in the expressionalgebraic and solve it in the same way as the expressions numerical. Therefore, it is necessary to know that the coefficient always multiplies the unknown that accompanies. As an example, let's calculate the numerical value of the expressionalgebraic then, knowing that x = 2 and y = 3.
4x2 + 5y
Substituting the numeric values of x and y in the expression, we have:
4·22 + 5·3
Note that the coefficient multiplies the unknown, but for ease of writing, the multiplication sign is omitted in the expressionsalgebraic. To finish solving, just calculate the resulting numeric expression:
4·22 + 5·3 = 4·4 + 5·3 = 16 + 15 = 31
It is worth mentioning that two unknowns that appear together are also being multiplied. If the expressionalgebraic above was:
2xy + xx + yy = 2xy + x2 + y2
Its numerical value would be:
2xy + x2 + y2 = 2·2·3 + 22 + 33 = 12 + 4 + 9 = 25
monomials
monomials they are expressionsalgebraic formed only by multiplying known numbers and incognitos. are examples of monomials:
1) 2x
2) 3x2y4
3) x
4) xy
5) 16
Realize that known numbers are considered monomials, as well as just the incognitos. In addition, the set of all unknowns and their exponents is called literal part, and the known number is called the coefficient of a monomium.
All basic math operations in monomials can be accomplished with some tweaking of rules and algorithms.
Addition and subtraction of monomials
Can only be performed when the monomials have partliteral identical. When this happens, add or subtract only the coefficients, keeping the literal part of the monomials in the final answer. For example:
2xy2k7 + 22xy2k7 – 20xy2k7 = 4xy2k7
For more information, details and examples on adding and subtracting monomials, Click here.
Multiplication and division of monomials
THE multiplication in monomials doesn't need the partsliterals are equal. To multiply two monomials, first multiply the coefficients and then multiply unknown by unknown using potency properties. For example:
4x3k2yz 15x2k4y = 60x3 + 2k2 + 4y1 + 1z = 60x5k6y2z
The division is done in the same way, however, the coefficients and use the power division property from the same basis to the literal part.
For more examples and details, see the text on splitting monomials. clicking here.
Polynomials
Polynomials are algebraic expressions formed by the algebraic addition of monomials. Thus, a polynomial is born when we add or subtract two distinct monomials. Heads up: every monomium is also a polynomial.
See some examples of polynomials:
1) 2x + 2x2
2) 2x + 3xy + 3y
3) 2ab + 16 - 4ab3
Addition and subtraction of polynomials
It is done by placing all similar terms side by side (monomials which have equal literal part) and adding them together. When the polynomials do not have similar terms, they cannot be added or subtracted. When polynomials have a term that is not similar to any other, that term is neither added nor subtracted, just repeated in the final result. For example:
(12x2 + 21y2 – 7k) + (– 15x2 + 25y2) =
12x2 + 21y2 – 7k – 15x2 + 25y2 =
12x2 – 15x2 + 21y2 + 25y2 – 7k =
– 3x2 + 46y2 – 7k
Polynomial Multiplication
THE multiplication in polynomials it is always done based on the distributive property of multiplication over addition (also known as a showerhead). Through it, we must multiply the first term of the first polynomial by all the terms of the second, then the second term of the first polynomial by all the terms of the second, and so on until all the terms of the first polynomial have been multiplied.
For that, of course, we use the power properties when necessary. For example:
(x2 + the2)(y2 + the2) = x2y2 + x2The2 + the2y2 + the4
More information and examples on multiplication, addition and subtraction of polynomials can be found clicking here.
polynomial division
It is the most difficult procedure of algebraic expressions. One of the most used techniques for sharepolynomials is very similar to the one used for dividing between real numbers: we look for a monomial that, multiplied by the highest-grade term of the divisor, equals the highest-grade term of the dividend. Then, just subtract the result of this multiplication from the dividend and “go down” the rest to continue the division. For example:
(x2 + 18x + 81): (x + 9) =
x2 + 18x + 81 | x + 9
– x2 – 9x x + 9
9x + 81
– 9x – 81
0
For more information on splitting polynomials and for more examples Click here.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-expressao-algebrica.htm