The concept of a polynomial
Definition of polynomial: A polynomial is the sum of monomials. Polynomial example:
here we see the sum of two monomials, and this is a polynomial, i.e. sum of monomials.
The terms that make up a polynomial are called terms of the polynomial.
Is the difference of monomials a polynomial? Yes, it is, because the difference is easily reduced to a sum, example: 5a – 2b = 5a + (-2b).
Monomials are also considered polynomials. But a monomial has no sum, then why is it considered a polynomial? And you can add zero to it and get its sum with a zero monomial. So the monomial is special case polynomial, it consists of one member.
The number zero is the zero polynomial.
Standard form of polynomial
What is a polynomial of standard form? A polynomial is the sum of monomials, and if all these monomials that make up the polynomial are written in standard form, and there should be no similar ones among them, then the polynomial is written in standard form.
An example of a polynomial in standard form:
here the polynomial consists of 2 monomials, each of which has standard view, there are no similar ones among the monomials.
Now an example of a polynomial that does not have a standard form:
here two monomials: 2a and 4a are similar. You need to add them up, then the polynomial will take the standard form:
Another example:
Is this polynomial reduced to standard form? No, his second term is not written in standard form. Writing it in standard form, we obtain a polynomial of standard form:
Polynomial degree
What is the degree of a polynomial?
Polynomial degree definition:
The degree of a polynomial is the highest degree that the monomials that make up a given polynomial of standard form have.
Example. What is the degree of the polynomial 5h? The degree of the polynomial 5h is equal to one, because this polynomial contains only one monomial and its degree is equal to one.
Another example. What is the degree of the polynomial 5a 2 h 3 s 4 +1? The degree of the polynomial 5a 2 h 3 s 4 + 1 is equal to nine, because this polynomial includes two monomials, the first monomial 5a 2 h 3 s 4 has the highest degree, and its degree is 9.
Another example. What is the degree of the polynomial 5? The degree of a polynomial 5 is zero. So, the degree of a polynomial consisting only of a number, i.e. without letters, equals zero.
The last example. What is the degree of the zero polynomial, i.e. zero? The degree of the zero polynomial is not defined.
A polynomial is the sum of monomials. If all the terms of a polynomial are written in standard form (see paragraph 51) and similar terms are reduced, you will get a polynomial of standard form.
Any integer expression can be converted into a polynomial of standard form - this is the purpose of transformations (simplifications) of integer expressions.
Let's look at examples in which an entire expression needs to be reduced to the standard form of a polynomial.
Solution. First, let's bring the terms of the polynomial to standard form. We obtain After bringing similar terms, we obtain a polynomial of the standard form
Solution. If there is a plus sign in front of the brackets, then the brackets can be omitted, preserving the signs of all terms enclosed in brackets. Using this rule for opening parentheses, we get:
Solution. If the parentheses are preceded by a minus sign, then the parentheses can be omitted by changing the signs of all terms enclosed in the brackets. Using this rule for hiding parentheses, we get:
Solution. The product of a monomial and a polynomial, according to the distributive law, is equal to the sum of the products of this monomial and each member of the polynomial. We get
Solution. We have
Solution. We have
It remains to give similar terms (they are underlined). We get:
53. Abbreviated multiplication formulas.
In some cases, bringing an entire expression to the standard form of a polynomial is carried out using the identities:
These identities are called abbreviated multiplication formulas,
Let's look at examples in which you need to convert a given expression into standard form myogochlea.
Example 1. .
Solution. Using formula (1), we obtain:
Example 2. .
Solution.
Example 3. .
Solution. Using formula (3), we obtain:
Example 4.
Solution. Using formula (4), we obtain:
54. Factoring polynomials.
Sometimes you can transform a polynomial into a product of several factors - polynomials or subnomials. Such an identity transformation is called factorization of a polynomial. In this case, the polynomial is said to be divisible by each of these factors.
Let's look at some ways to factor polynomials,
1) Taking the common factor out of brackets. This transformation is a direct consequence of the distributive law (for clarity, you just need to rewrite this law “from right to left”):
Example 1: Factor a polynomial
Solution. .
Usually, when taking the common factor out of brackets, each variable included in all terms of the polynomial is taken out with the lowest exponent that it has in this polynomial. If all the coefficients of the polynomial are integers, then the largest absolute common divisor of all coefficients of the polynomial is taken as the coefficient of the common factor.
2) Using abbreviated multiplication formulas. Formulas (1) - (7) from paragraph 53, being read from right to left, in many cases turn out to be useful for factoring polynomials.
Example 2: Factor .
Solution. We have. Applying formula (1) (difference of squares), we obtain . By applying
Now formulas (4) and (5) (sum of cubes, difference of cubes), we get:
Example 3. .
Solution. First, let's take the common factor out of the bracket. To do this, we will find the greatest common divisor of the coefficients 4, 16, 16 and the smallest exponents with which the variables a and b are included in the constituent monomials of this polynomial. We get:
3) Method of grouping. It is based on the fact that the commutative and associative laws of addition allow the members of a polynomial to be grouped different ways. Sometimes it is possible to group in such a way that after taking the common factors out of brackets, the same polynomial remains in brackets in each group, which in turn, as a common factor, can be taken out of brackets. Let's look at examples of factoring a polynomial.
Example 4. .
Solution. Let's do the grouping as follows:
In the first group, let's take the common factor out of the brackets into the second - the common factor 5. We get Now we put the polynomial as a common factor out of the brackets: Thus, we get:
Example 5.
Solution. .
Example 6.
Solution. Here, no grouping will lead to the appearance of the same polynomial in all groups. In such cases, it is sometimes useful to represent a member of the polynomial as a sum, and then try the grouping method again. In our example, it is advisable to represent it as a sum. We get
Example 7.
Solution. Add and subtract a monomial We get
55. Polynomials in one variable.
A polynomial, where a, b are variable numbers, is called a polynomial of the first degree; a polynomial where a, b, c are variable numbers, called a polynomial of the second degree or a square trinomial; a polynomial where a, b, c, d are numbers, the variable is called a polynomial of the third degree.
In general, if o is a variable, then it is a polynomial
called lsmogochnolenol degree (relative to x); , m-terms of the polynomial, coefficients, the leading term of the polynomial, a is the coefficient of the leading term, the free term of the polynomial. Typically, a polynomial is written in descending powers of a variable, i.e., the powers of a variable gradually decrease, in particular, the leading term is in first place, and the free term is in last place. The degree of a polynomial is the degree of the highest term.
For example, a polynomial of the fifth degree, in which the leading term, 1, is the free term of the polynomial.
The root of a polynomial is the value at which the polynomial vanishes. For example, the number 2 is the root of a polynomial since
Among the various expressions that are considered in algebra are important place occupy sums of monomials. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8\)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2\)
The sum of monomials is called a polynomial. The terms in a polynomial are called terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
For example, a polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
Let us represent all terms in the form of monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
Let us present similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all terms of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Behind degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1\)
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
Using the distributive property of multiplication, you can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)
The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
Product of polynomials. Transformation (simplification) of the product of two polynomials
In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.
Usually the following rule is used.
To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.
Abbreviated multiplication formulas. Sum squares, differences and difference of squares
You have to deal with some expressions in algebraic transformations more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.
\((a + b)^2 = a^2 + b^2 + 2ab \) - square of the sum equal to the sum squares and double the product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the doubled product.
\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.
These three identities allow one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.
§ 1 What is a polynomial?
In this lesson we will learn what mathematicians call a polynomial and which polynomial is a polynomial of standard form.
Very often, when solving real problems, we come across algebraic expressions that contain the sum of dissimilar monomials. It is impossible to add such monomials, but the situation is not so hopeless. To work with such sums, mathematicians introduced a new term “polynomial”. Let's give a definition.
A polynomial is the sum of several monomials.
For example, expressions
The monomials included in a polynomial are called terms of the polynomial. The number of terms of a polynomial can be any.
For some polynomials, the specific names binomial and trinomial are often used. This means that the polynomial consists of two or three monomials.
For example:
In mathematics, polynomials are also called polynomials. This word comes from the Greek words poly, which means “many”, and the word nomos, which means “part”. And the first letter of the word poly is used to denote polynomials.
To do this, write down the letter p and next to it in parentheses, separated by a semicolon, list those variables that are part of the polynomial.
The entry p(x) is read as “pe from x”, the entry p(x;y) is read as “pe from x, igrek”, etc. Then they put an equal sign and write the polynomial itself.
For example:
This form of notation is convenient when finding the value of a polynomial. The value of a polynomial is the value of an algebraic expression given the value of the letters.
For example, given a polynomial:
Need to find:
This task should be understood as follows: you need to find the value of the expression 2x-3 at x=5.
Substitute the number 5 instead of x, we get
Or this example:
This task should be understood as follows:
We substitute these values and get:
§ 2 Polynomial of standard form
This is, of course, a polynomial, only the monomials included in it are written in a non-standard form. Let us reduce all monomials to standard form.
But that's not all. We see that the first and second monomials are similar. Therefore, similar terms can be given.
Nothing more can be done. We have obtained a polynomial equal to the original one, but all its monomials are written in standard form, and similar terms are given.
Such a polynomial is called a polynomial of standard form.
Any polynomial can be reduced to standard form, and this procedure must be performed first before performing any operations with polynomials.
Let's look at another example.
This polynomial consists of five monomials, and not all of them are written in standard form.
To bring them to standard form:
But this is not enough. We also need to give similar monomials.
In this polynomial, all monomials are written in standard form, and all similar terms are given, which means it is a polynomial of standard form.
Thus, today we became acquainted with the new mathematical concept of a polynomial, learned to write it in standard form and find the value of the polynomial.
List of used literature:
- Mordkovich A.G., Algebra 7th grade in 2 parts, Part 1, Textbook for educational institutions/ A.G. Mordkovich. – 10th ed., revised – Moscow, “Mnemosyne”, 2007
- Mordkovich A.G., Algebra 7th grade in 2 parts, Part 2, Problem book for educational institutions / [A.G. Mordkovich and others]; edited by A.G. Mordkovich - 10th edition, revised - Moscow, “Mnemosyne”, 2007
- HER. Tulchinskaya, Algebra 7th grade. Blitz survey: a manual for students of general education institutions, 4th edition, revised and expanded, Moscow, “Mnemosyne”, 2008
- Alexandrova L.A., Algebra 7th grade. Thematic test work in new form for students of general education institutions, edited by A.G. Mordkovich, Moscow, “Mnemosyne”, 2011
- Alexandrova L.A. Algebra 7th grade. Independent work for students of general education institutions, edited by A.G. Mordkovich - 6th edition, stereotypical, Moscow, “Mnemosyne”, 2010
After studying monomials, we move on to polynomials. This article will tell you about all the necessary information required to perform actions on them. We will define a polynomial with accompanying definitions term of a polynomial, that is, free and similar, consider a polynomial of a standard form, introduce a degree and learn how to find it, work with its coefficients.
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Polynomial and its terms - definitions and examples
The definition of a polynomial was necessary back in 7 class after studying monomials. Let's look at its full definition.
Definition 1
Polynomial The sum of monomials is calculated, and the monomial itself is a special case of a polynomial.
From the definition it follows that examples of polynomials can be different: 5 , 0 , − 1 , x, 5 a b 3, x 2 · 0 , 6 · x · (− 2) · y 12 , - 2 13 · x · y 2 · 3 2 3 · x · x 3 · y · z and so on. From the definition we have that 1+x, a 2 + b 2 and the expression x 2 - 2 x y + 2 5 x 2 + y 2 + 5, 2 y x are polynomials.
Let's look at some more definitions.
Definition 2
Members of the polynomial its constituent monomials are called.
Consider an example where we have a polynomial 3 x 4 − 2 x y + 3 − y 3, consisting of 4 terms: 3 x 4, − 2 x y, 3 and − y 3. Such a monomial can be considered a polynomial, which consists of one term.
Definition 3
Polynomials that contain 2, 3 trinomials have the corresponding name - binomial And trinomial.
It follows that an expression of the form x+y– is a binomial, and the expression 2 x 3 q − q x x x + 7 b is a trinomial.
By school curriculum worked with a linear binomial of the form a · x + b, where a and b are some numbers, and x is a variable. Let's consider examples of linear binomials of the form: x + 1, x · 7, 2 − 4 with examples of square trinomials x 2 + 3 · x − 5 and 2 5 · x 2 - 3 x + 11.
To transform and solve, it is necessary to find and bring similar terms. For example, a polynomial of the form 1 + 5 x − 3 + y + 2 x has similar terms 1 and - 3, 5 x and 2 x. They are divided into special group called similar terms of a polynomial.
Definition 4
Similar terms of a polynomial are similar terms found in a polynomial.
In the example above, we have that 1 and - 3, 5 x and 2 x are similar terms of the polynomial or similar terms. In order to simplify the expression, find and reduce similar terms.
Polynomial of standard form
All monomials and polynomials have their own specific names.
Definition 5
Polynomial of standard form is a polynomial in which each term included in it has a monomial of standard form and does not contain similar terms.
From the definition it is clear that it is possible to reduce polynomials of the standard form, for example, 3 x 2 − x y + 1 and __formula__, and the entry is in standard form. The expressions 5 + 3 · x 2 − x 2 + 2 · x · z and 5 + 3 · x 2 − x 2 + 2 · x · z are not polynomials of standard form, since the first of them has similar terms in the form 3 · x 2 and − x 2, and the second contains a monomial of the form x · y 3 · x · z 2, which differs from the standard polynomial.
If circumstances require it, sometimes the polynomial is reduced to a standard form. The concept of a free term of a polynomial is also considered a polynomial of standard form.
Definition 6
Free term of a polynomial is a polynomial of standard form that does not have a literal part.
In other words, when a polynomial in standard form has a number, it is called a free member. Then the number 5 is the free term of the polynomial x 2 z + 5, and the polynomial 7 a + 4 a b + b 3 does not have a free term.
Degree of a polynomial - how to find it?
The definition of the degree of a polynomial itself is based on the definition of a standard form polynomial and on the degrees of the monomials that are its components.
Definition 7
Degree of a polynomial of standard form is called the largest of the degrees included in its notation.
Let's look at an example. The degree of the polynomial 5 x 3 − 4 is equal to 3, because the monomials included in its composition have degrees 3 and 0, and the larger of them is 3, respectively. The definition of the degree from the polynomial 4 x 2 y 3 − 5 x 4 y + 6 x is equal to the largest of the numbers, that is, 2 + 3 = 5, 4 + 1 = 5 and 1, which means 5.
It is necessary to find out how the degree itself is found.
Definition 8
Degree of a polynomial of an arbitrary number is the degree of the corresponding polynomial in standard form.
When a polynomial is not written in standard form, but you need to find its degree, you need to reduce it to the standard form, and then find the required degree.
Example 1
Find the degree of a polynomial 3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12.
Solution
First, let's present the polynomial in standard form. We get an expression of the form:
3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12 = = (3 a 12 − 2 a 12 − a 12) − 2 · (a · a) · (b · b) · (c · c) + y 2 · z 2 = = − 2 · a 2 · b 2 · c 2 + y 2 · z 2
When obtaining a polynomial of standard form, we find that two of them stand out clearly - 2 · a 2 · b 2 · c 2 and y 2 · z 2 . To find the degrees, we count and find that 2 + 2 + 2 = 6 and 2 + 2 = 4. It can be seen that the largest of them is 6. From the definition it follows that 6 is the degree of the polynomial − 2 · a 2 · b 2 · c 2 + y 2 · z 2 , and therefore the original value.
Answer: 6 .
Coefficients of polynomial terms
Definition 9When all the terms of a polynomial are monomials of the standard form, then in this case they have the name coefficients of polynomial terms. In other words, they can be called coefficients of the polynomial.
When considering the example, it is clear that a polynomial of the form 2 x − 0, 5 x y + 3 x + 7 contains 4 polynomials: 2 x, − 0, 5 x y, 3 x and 7 with their corresponding coefficients 2, − 0, 5, 3 and 7. This means that 2, − 0, 5, 3 and 7 are considered coefficients of terms of a given polynomial of the form 2 x − 0, 5 x y + 3 x + 7. When converting, it is important to pay attention to the coefficients in front of the variables.
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