We said that they take place as polynomials standard view, and not standard. There we noted that anyone can bring the polynomial to standard form. In this article, we will first find out what meaning this phrase carries. Next we list the steps to convert any polynomial into standard form. Finally, let's look at solutions to typical examples. We will describe the solutions in great detail in order to understand all the nuances that arise when reducing polynomials to standard form.
Page navigation.
What does it mean to reduce a polynomial to standard form?
First you need to clearly understand what is meant by reducing a polynomial to standard form. Let's figure this out.
Polynomials, like any other expressions, can be subjected to identical transformations. As a result of performing such transformations, expressions are obtained that are identically equal to the original expression. Thus, performing certain transformations with polynomials of non-standard form allows one to move on to polynomials that are identically equal to them, but written in standard form. This transition is called reducing the polynomial to standard form.
So, reduce the polynomial to standard form- this means replacing the original polynomial with an identically equal polynomial of a standard form, obtained from the original one by carrying out identical transformations.
How to reduce a polynomial to standard form?
Let's think about what transformations will help us bring the polynomial to a standard form. We will start from the definition of a polynomial of the standard form.
By definition, every term of a polynomial of standard form is a monomial of standard form, and a polynomial of standard form contains no similar terms. In turn, polynomials written in a form other than the standard one can consist of monomials in a non-standard form and can contain similar terms. This logically follows next rule, explaining how to reduce a polynomial to standard form:
- first you need to bring the monomials that make up the original polynomial to standard form,
- then perform the reduction of similar terms.
As a result, a polynomial of standard form will be obtained, since all its terms will be written in standard form, and it will not contain similar terms.
Examples, solutions
Let's look at examples of reducing polynomials to standard form. When solving, we will follow the steps dictated by the rule from the previous paragraph.
Here we note that sometimes all the terms of a polynomial are immediately written in standard form; in this case, it is enough to just give similar terms. Sometimes, after reducing the terms of a polynomial to a standard form, there are no similar terms, therefore, the stage of bringing similar terms is omitted in this case. IN general case you have to do both.
Example.
Present the polynomials in standard form: 5 x 2 y+2 y 3 −x y+1 , 0.8+2 a 3 0.6−b a b 4 b 5 And .
Solution.
All terms of the polynomial 5·x 2 ·y+2·y 3 −x·y+1 are written in standard form; it does not have similar terms, therefore, this polynomial is already presented in standard form.
Let's move on to the next polynomial 0.8+2 a 3 0.6−b a b 4 b 5. Its form is not standard, as evidenced by the terms 2·a 3 ·0.6 and −b·a·b 4 ·b 5 of a non-standard form. Let's present it in standard form.
At the first stage of bringing the original polynomial to standard form, we need to present all its terms in standard form. Therefore, we bring the monomial 2·a 3 ·0.6 to the standard form, we have 2·a 3 ·0.6=1.2·a 3 , after which – the monomial −b·a·b 4 ·b 5 , we have −b·a·b 4 ·b 5 =−a·b 1+4+5 =−a·b 10. Thus, . In the resulting polynomial, all terms are written in standard form; moreover, it is obvious that there are no similar terms in it. Consequently, this completes the reduction of the original polynomial to standard form.
It remains to present the last of the given polynomials in standard form. After bringing all its members to standard form, it will be written as 
. It has similar members, so you need to cast similar members: 
So the original polynomial took the standard form −x·y+1.
Answer:
5 x 2 y+2 y 3 −x y+1 – already in standard form, 0.8+2 a 3 0.6−b a b 4 b 5 =0.8+1.2 a 3 −a b 10, .
Often, bringing a polynomial to a standard form is only an intermediate step in answering the question posed to the problem. For example, finding the degree of a polynomial requires its preliminary representation in standard form.
Example.
Give a polynomial 
to the standard form, indicate its degree and arrange the terms in descending degrees of the variable.
Solution.
First, we bring all the terms of the polynomial to standard form: 
.
Now we present similar terms: 
So we brought the original polynomial to a standard form, this allows us to determine the degree of the polynomial, which is equal to the highest degree of the monomials included in it. Obviously it is equal to 5.
It remains to arrange the terms of the polynomial in decreasing powers of the variables. To do this, you just need to rearrange the terms in the resulting polynomial of standard form, taking into account the requirement. The term z 5 has the highest degree; the degrees of the terms , −0.5·z 2 and 11 are equal to 3, 2 and 0, respectively. Therefore, a polynomial with terms arranged in decreasing powers of the variable will have the form 
.
Answer:
The degree of the polynomial is 5, and after arranging its terms in descending degrees of the variable, it takes the form .
References.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
- Algebra and the beginning of mathematical analysis. 10th grade: textbook. for general education institutions: basic and profile. levels / [Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; edited by A. B. Zhizhchenko. - 3rd ed. - M.: Education, 2010.- 368 p. : ill. - ISBN 978-5-09-022771-1.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. 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.
For 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
With some expressions in algebraic transformations have to deal with 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.
\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double 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 in transformations to replace their left parts with right ones and vice versa - right parts with left 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.
Polynomial and its standard form
A polynomial is the sum of monomials.
The monomials that make up a polynomial are called members of the polynomial. So the terms of the polynomial 4x2y - 5xy + 3x -1 are 4x2y, -5xy, 3x and -1.
If a polynomial consists of two terms, then it is called a binomial, if it consists of three, it is called a trinomial. A monomial is considered a polynomial consisting of one term.
In the polynomial 7x3y2 - 12 + 4x2y - 2y2x3 + 6, the terms 7x3y2 and - 2y2x3 are similar terms because they have the same letter part. The terms -12 and 6, which do not have a letter part, are also similar. Similar terms in a polynomial are called similar terms of a polynomial, and the reduction of similar terms in a polynomial is called a reduction of similar terms of a polynomial.
As an example, let us give similar terms in the polynomial 7x3y2 - 12 + 4x2y - 2y2x3 + 6 = 5x3y2 + 4x2y - 6.
A polynomial is called a polynomial of standard form if each of its terms is a monomial of standard form and this polynomial does not contain similar terms.
Any polynomial can be reduced to standard form. To do this, you need to present each of its members in standard form and bring similar terms.
The degree of a polynomial of standard form is the greatest of the degrees of its constituent monomials.
The degree of an arbitrary polynomial is the degree of an identically equal polynomial of standard form.
For example, let's find the degree of the polynomial 8x4y2 - 12 + 4x2y - 3y2x4 + 6 - 5y2x4:
8x4y2 - 12 + 4x2y - 3y2x4 + 6 - 5y2x4 = 4x2y -6.
Note that the original polynomial includes monomials of the sixth degree, but when similar terms were reduced, all of them were reduced, and the result was a polynomial of the third degree, which means the original polynomial has degree 3!
Polynomials in one variable
An expression of the form where are some numbers and is called a polynomial of degree from.
Two polynomials are said to be identically equal if they numeric values coincide for all values. Polynomials and are identically equal if and only if they coincide, i.e. coefficients for equal degrees these polynomials are the same.
When dividing a polynomial by a polynomial (for example, by a “corner”), we obtain a polynomial (incomplete quotient) and a remainder - a polynomial (in the case when the remainder is zero, the polynomial is called a quotient). If is the dividend and is the divisor, then we represent the polynomial in the form. In this case, the sum of the degrees of polynomials is equal to the degree of the polynomial, and the degree of the remainder is less than the degree of the divisor.
The concept of a polynomial. Polynomial degree
A polynomial in the variable x is an expression of the form
anxn+an-1xn-1+... +a1x+a0,where n - natural number; аn, an-1,..., a1, a0 - any numbers called the coefficients of this polynomial. The expressions anxn, an-1xn-1,..., a1х, a0 are called terms of the polynomial, a0 is the free term.
We will often use the following terms: an - coefficient for xn, an-1 - coefficient for xn-1, etc.
Examples of polynomials are the following expressions: 0x4+2x3+ (-3) x3+ (3/7) x+; 0x2+0x+3; 0x2+0x+0. Here, for the first polynomial, the coefficients are the numbers 0, 2, - 3, 3/7, ; in this case, for example, the number 2 is the coefficient of x3, and is the free term.
A polynomial whose coefficients are all zero is called zero.
So, for example, the polynomial 0x2+0x+0 is zero.
From the notation of a polynomial it is clear that it consists of several members. This is where the term ‹‹polynomial›› (many terms) comes from. Sometimes a polynomial is called a polynomial. This term comes from Greek wordsπολι - many and νομχ - member.
We will denote a polynomial in one variable x as follows: f (x), g (x), h (x), etc. for example, if the first of the above polynomials is denoted by f (x), then we can write: f (x) =0x4+2x3+ (-3) x2+3/7x+.
In order to make the polynomial notation simpler and more compact, we agreed on a number of conventions.
Those terms of a non-zero polynomial whose coefficients are equal to zero are not written down. For example, instead of f (x) =0x3+3x2+0x+5 they write: f (x) =3x2+5; instead of g (x) =0x2+0x+3 - g (x) =3. Thus, every number is also a polynomial. A polynomial h (x) for which all coefficients are equal to zero, i.e. zero polynomial is written as follows: h (x) =0.
Coefficients of a polynomial that are not a free member and equal to 1 are also not written down. For example, the polynomial f (x) =2x3+1x2+7x+1 can be written as follows: f (x) =x3+x2+7x+1.
The sign ‹‹-›› of a negative coefficient is assigned to the term containing this coefficient, i.e., for example, the polynomial f (x) =2x3+ (-3) x2+7x+ (-5) is written as f (x) =2x3 -3x2+7x-5. Moreover, if the coefficient, which is not a free term, is equal to - 1, then the “-” sign is kept in front of the corresponding term, and the unit is not written. For example, if the polynomial has the form f (x) =x3+ (-1) x2+3x+ (-1), then it can be written as follows: f (x) =x3-x2+3x-1.
The question may arise: why, for example, agree to replace 1x with x in the notation of a polynomial if it is known that 1x = x for any number x? The point is that the last equality holds if x is a number. In our case, x is an element of arbitrary nature. Moreover, we do not yet have the right to consider the entry 1x as the product of the number 1 and the element x, because, we repeat, x is not a number. It is precisely this circumstance that causes the conventions in writing a polynomial. And if we continue to talk about the product of, say, 2 and x without any reason, then we are admitting some lack of rigor.
Due to conventions in writing a polynomial, we pay attention to this detail. If there is, for example, a polynomial f (x) = 3x3-2x2-x+2, then its coefficients are the numbers 3, - 2, - 1.2. Of course, one could say that the coefficients are the numbers 0, 3, - 2, - 1, 2, meaning this representation of this polynomial: f (x) = 0x4-3x2-2x2-x+2.
In the future, for definiteness, we will indicate the coefficients, starting with non-zero ones, in the order they appear in the notation of the polynomial. Thus, the coefficients of the polynomial f (x) = 2x5-x are the numbers 2, 0, 0, 0, - 1, 0. The fact is that although, for example, the term with x2 is absent in the notation, this only means that its coefficient equal to zero. Similarly, there is no free term in the entry, since it is equal to zero.
If there is a polynomial f (x) =anxn+an-1xn-1+... +a1x+a0 and an≠0, then the number n is called the degree of the polynomial f (x) (or they say: f (x) - nth degree) and write Art. f(x)=n. In this case, an is called the leading coefficient, and anxn is the leading term of this polynomial.
For example, if f (x) =5x4-2x+3, then art. f (x) =4, leading coefficient - 5, leading term - 5x4.
Let us now consider the polynomial f (x) =a, where a is a non-zero number. What is the degree of this polynomial? It is easy to see that the coefficients of the polynomial f (x) =anxn+an-1xn-1+... +a1x+a0 are numbered from right to left with the numbers 0, 1, 2, …, n-1, n and if an≠0, then Art. f(x)=n. This means that the degree of a polynomial is the largest of the numbers of its coefficients that are different from zero (with the numbering that was just mentioned). Let us now return to the polynomial f (x) =a, a≠0, and number its coefficients from right to left with the numbers 0, 1, 2, ... coefficient a will receive the number 0, and since all other coefficients are zero, then this is the largest non-zero coefficient number of a given polynomial. So Art. f (x) =0.
Thus, polynomials of degree zero are numbers other than zero.
It remains to find out what the situation is with the degree of the zero polynomial. As is known, all its coefficients are equal to zero, and therefore the above definition cannot be applied to it. So, we agreed not to assign any degree to the zero polynomial, i.e. that he doesn't have a degree. This convention is caused by some circumstances that will be discussed a little later.
So, the zero polynomial has no degree; the polynomial f (x) =a, where a is a non-zero number and has degree 0; the degree of any other polynomial, as is easy to see, is equal to the highest rate power of the variable x, the coefficient of which is equal to zero.
In conclusion, let us recall a few more definitions. A polynomial of the second degree f (x) =ax2+bx+c is called a square trinomial. A polynomial of the first degree of the form g (x) =x+c is called a linear binomial.
Horner's scheme.
Horner's scheme is one of the simplest ways to divide a polynomial by a binomial x-a. Of course, the application of Horner’s scheme is not limited to division, but first let’s consider just that. We will explain the use of the algorithm with examples. Divide by. Let's make a table of two lines: in the first line we write the coefficients of the polynomial in descending order of degrees of the variable. Note that this polynomial does not contain x, i.e. the coefficient in front of x is 0. Since we are dividing by, we write one in the second line:
Let's start filling in the empty cells in the second line. Let's write 5 into the first empty cell, simply moving it from the corresponding cell of the first row:
Let's fill the next cell according to this principle:
Let's fill in the fourth one in the same way:
For the fifth cell we get:
And finally, for the last, sixth cell, we have:
The problem is solved, all that remains is to write down the answer:
As you can see, the numbers located in the second line (between the first and last) are the coefficients of the polynomial obtained after dividing by. The last number in the second line means the remainder of the division or, which is the same, the value of the polynomial at. Consequently, if in our case the remainder is equal to zero, then the polynomials are divided entirely.
The result also indicates that 1 is the root of the polynomial.
Let's give another example. Let's divide the polynomial by. Let us immediately stipulate that the expression must be presented in the form. Horner’s scheme will involve exactly -3.
If our goal is to find all the roots of a polynomial, then Horner's scheme can be applied several times in a row until we have exhausted all the roots. For example, let's find all the roots of a polynomial. Whole roots must be looked for among the divisors of the free term, i.e. among the divisors there are 8. That is, the numbers -8, -4, -2, -1, 1, 2, 4, 8 can be integer roots. Let's check, for example, 1:
So, the remainder is 0, i.e. unity is indeed the root of this polynomial. Let's try to check the unit a few more times. We will not create a new table for this, but will continue to use the previous one:
Again the remainder is zero. Let's continue the table until we've exhausted everything. possible values roots:
Bottom line: Of course this method selection is ineffective in the general case, when the roots are not integers, but for integer roots the method is quite good.
RATIONAL ROOTS OF A POLYNOMIAL WITH INTEGER COEFFICIENTS Finding the roots of a polynomial is an interesting and rather difficult problem, the solution of which goes beyond the limits of school course mathematics. However, for polynomials with integer coefficients there is a simple enumeration algorithm that allows you to find all rational roots.
Theorem. If a polynomial with integer coefficients has a rational root (is an irreducible fraction),
then the numerator of the fraction is the divisor of the free term, and the denominator is the divisor of the leading coefficient of this polynomial.
Proof
Let the polynomial be written in canonical form. Let us substitute and get rid of the denominators by multiplying by the largest power n:
Move the member to the right
The product is divided by the integer m. By condition, the fraction is irreducible, therefore, the numbers m and n are coprime. Then the numbers m will be coprime and If the product of numbers is divisible by m, and the factor is coprime by m, then the second factor must be divisible by m.
The proof of the divisibility of the leading coefficient by the denominator n is proved in the same way, moving the term to the right and moving the factor n out of the left bracket from the left.
Let us make a few comments to the proven theorem.
Notes
1) The theorem gives only necessary condition existence of a rational root. This means you need to check everything rational numbers, with the property specified in the theorem and select from them those that turn out to be roots. There will be no others.
2) Among the divisors, you must take not only positive, but also negative integers.
3) If the leading coefficient is 1, then every rational root must be an integer, since 1 has no divisors except
Let us illustrate the theorem and comments to it with examples.
1) Rational roots must be whole.
We sort out the divisors of the free term: Positive numbers there is no point in substituting, since all the coefficients of the polynomial are positive and at
It remains to calculate F(–1) and F(–2). F(–1)=1+0; F(–2)=0.
So, the polynomial has one integer root x=–2.
We can divide F(x) by x+2:
2) Write down the possible values of the roots:
By substitution we are convinced that the Polynomial also has three different rational roots:
Of course, the root x = -1 is easy to guess. Then you can factorize and look for the roots of the quadratic trinomial using the usual techniques.
DIVISION OF POLYNOMIALS. EUCLID ALGORITHM
Division of polynomials
The result of division is a single pair of polynomials - the quotient and the remainder, which must satisfy the equality:< делимое > = < делитель > ´ < частное > + <… Если многочлен степени n Pn(x) является делимым,Example No. 1
6x 3 + x 2 – 3x – 2 2x 2 – x – 1
6x 3 ± 3x 2 ± 3x 3x + 2
4x 2 + 0x – 2
4x 2 ± 2x ± 2
Thus, 6x 3 + x 2 – 3x – 2 = (2x 2 – x – 1)(3x + 2) + 2x.
Example No. 2
a 5 a 4 b a 4 –a 3 b + a 2 b 2 – ab 3 + b 4
± a 4 b ± a 3 b 2
– a 2 b 3 + b 5
± a 2 b 3 ± ab 4
Thus, a 5 + b 5 = (a + b)(a 4 –a 3 b + a 2 b 2 – ab 3 + b 4).
Lesson on the topic: "The concept and definition of a polynomial. Standard form of a polynomial"
Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an anti-virus program.
Teaching aids and simulators in the Integral online store for grade 7
Electronic textbook based on the textbook by Yu.N. Makarycheva
Electronic textbook based on the textbook by Sh.A. Alimova
Guys, you have already studied monomials in the topic: Standard form of a monomial. Definitions. Examples. Let's review the basic definitions.
Monomial– an expression consisting of a product of numbers and variables. Variables can be raised to natural powers. A monomial does not contain any operations other than multiplication.
Standard form of monomial- this type when the coefficient (numerical factor) comes first, followed by the degrees of various variables.
Similar monomials– these are either identical monomials, or monomials that differ from each other by a coefficient.
The concept of a polynomial
A polynomial, like a monomial, is a generalized name for mathematical expressions of a certain type. We have encountered such generalizations before. For example, “sum”, “product”, “exponentiation”. When we hear “number difference,” the thought of multiplication or division does not even occur to us. Also, a polynomial is an expression of a strictly defined type.Definition of a polynomial
Polynomial is the sum of the monomials.The monomials that make up a polynomial are called members of the polynomial. If there are two terms, then we are dealing with a binomial, if there are three, then with a trinomial. If there are more terms, it is a polynomial.
Examples of polynomials.
1) 2аb + 4сd (binomial);
2) 4ab + 3cd + 4x (trinomial);
3) 4a 2 b 4 + 4c 8 d 9 + 2xу 3 ;
3c 7 d 8 - 2b 6 c 2 d + 7xy - 5xy 2.
Let's look carefully at the last expression. By definition, a polynomial is the sum of monomials, but in the last example we not only add, but also subtract monomials.
To clarify, let's look at a small example.
Let's write down the expression a + b - c(let's agree that a ≥ 0, b ≥ 0 and c ≥0) and answer the question: is this the sum or the difference? It's hard to say.
Indeed, if we rewrite the expression as a + b + (-c), we get the sum of two positive and one negative terms.
If you look at our example, we are dealing specifically with the sum of monomials with coefficients: 3, - 2, 7, -5. In mathematics there is a term "algebraic sum". Thus, in the definition of a polynomial we mean an “algebraic sum”.
But a notation of the form 3a: b + 7c is not a polynomial because 3a: b is not a monomial.
The notation of the form 3b + 2a * (c 2 + d) is also not a polynomial, since 2a * (c 2 + d) is not a monomial. If you open the brackets, the resulting expression will be a polynomial.
3b + 2a * (c 2 + d) = 3b + 2ac 2 + 2ad.
Polynomial degree is the highest degree of its members.
The polynomial a 3 b 2 + a 4 has the fifth degree, since the degree of the monomial a 3 b 2 is 2 + 3= 5, and the degree of the monomial a 4 is 4.
Standard form of polynomial
A polynomial that does not have similar terms and is written in descending order of the powers of the terms of the polynomial is a polynomial of standard form.The polynomial is brought to a standard form in order to remove unnecessary cumbersome writing and simplify further actions with it.
Indeed, why, for example, write the long expression 2b 2 + 3b 2 + 4b 2 + 2a 2 + a 2 + 4 + 4, when it can be written shorter than 9b 2 + 3a 2 + 8.
To bring a polynomial to standard form, you need to:
1. bring all its members to a standard form,
2. add similar (identical or with different numerical coefficients) terms. This procedure is often called bringing similar.
Example.
Reduce the polynomial aba + 2y 2 x 4 x + y 2 x 3 x 2 + 4 + 10a 2 b + 10 to standard form.
Solution.
a 2 b + 2 x 5 y 2 + x 5 y 2 + 10a 2 b + 14= 11a 2 b + 3 x 5 y 2 + 14.
Let's determine the powers of the monomials included in the expression and arrange them in descending order.
11a 2 b has the third degree, 3 x 5 y 2 has the seventh degree, 14 has the zero degree.
This means that we will put 3 x 5 y 2 (7th degree) in first place, 12a 2 b (3rd degree) in second place, and 14 (zero degree) in third place.
As a result, we obtain a polynomial of the standard form 3x 5 y 2 + 11a 2 b + 14.
Examples for self-solution
Reduce polynomials to standard form.1) 4b 3 aa - 5x 2 y + 6ac - 2b 3 a 2 - 56 + ac + x 2 y + 50 * (2 a 2 b 3 - 4x 2 y + 7ac - 6);
2) 6a 5 b + 3x 2 y + 45 + x 2 y + ab - 40 * (6a 5 b + 4xy + ab + 5);
3) 4ax 2 + 5bc - 6a - 24bc + xx 4 x (5ax 6 - 19bc - 6a);
4) 7abc 2 + 5acbc + 7ab 2 - 6bab + 2cabc (14abc 2 + ab 2).
In this lesson, we will recall the basic definitions of this topic and consider some typical problems, namely, reducing a polynomial to a standard form and calculating a numerical value for given values of variables. We will solve several examples in which reduction to a standard form will be used to solve various kinds of problems.
Subject:Polynomials. Arithmetic operations on monomials
Lesson:Reducing a polynomial to standard form. Typical tasks
Let us recall the basic definition: a polynomial is the sum of monomials. Each monomial that is part of a polynomial as a term is called its member. For example:
Binomial;
Polynomial;
Binomial;
Since a polynomial consists of monomials, the first action with a polynomial follows from here - you need to bring all monomials to a standard form. Let us remind you that to do this you need to multiply all the numerical factors - get a numerical coefficient, and multiply the corresponding powers - get the letter part. In addition, let us pay attention to the theorem about the product of powers: when multiplying powers, their exponents add up.
Let's consider an important operation - reducing a polynomial to standard form. Example:
Comment: to bring a polynomial to a standard form, you need to bring all the monomials included in its composition to a standard form, after which, if there are similar monomials - and these are monomials with the same letter part - perform actions with them.
So, we looked at the first typical problem - bringing a polynomial to a standard form.
The next typical problem is calculating the specific value of a polynomial for given numerical values of the variables included in it. Let's continue to look at the previous example and set the values of the variables:
Comment: let us recall that one to any natural power is equal to one, and zero to any natural power is equal to zero, in addition, we recall that when multiplying any number by zero, we get zero.
Let's look at a number of examples of typical operations of reducing a polynomial to a standard form and calculating its value:
Example 1 - bring to standard form:
Comment: the first step is to bring the monomials to standard form, you need to bring the first, second and sixth; second action - we bring similar terms, that is, we perform the given arithmetic operations on them: we add the first with the fifth, the second with the third, we rewrite the rest without changes, since they have no similar ones.
Example 2 - calculate the value of the polynomial from example 1 given the values of the variables:
Comment: when calculating, you should remember that a unit to any natural power is one; if it is difficult to calculate powers of two, you can use the table of powers.
Example 3 - instead of an asterisk, put a monomial such that the result does not contain a variable:
Comment: regardless of the task, the first action is always the same - bring the polynomial to a standard form. In our example, this action comes down to bringing similar terms. After this, you should carefully read the condition again and think about how we can get rid of the monomial. Obviously, to do this, you need to add the same monomial to it, but with the opposite sign - . Next, we replace the asterisk with this monomial and make sure that our solution is correct.
