A polynomial is the sum of monomials, that is, products of numbers and variables. It is more convenient to work with it, since most often converting an expression into a polynomial allows you to significantly simplify it.
Instructions
Expand all parentheses of the expression. To do this, use formulas, for example, (a+b)^2=a^2+2ab+b^2. If you do not know the formulas, or they are difficult to apply to a given expression, open the brackets sequentially. To do this, multiply the first term of the first expression by each term of the second expression, then the second term of the first expression by each term of the second, etc. As a result, all elements of both brackets will be multiplied together.
If you have three expressions in parentheses, first multiply the first two, leaving the third expression untouched. After simplifying the result obtained by transforming the first parentheses, multiply it with the third expression.
Carefully follow the signs in front of monomial factors. If you multiply two terms with the same sign (for example, both are positive or both are negative), the monomial will have a "+" sign. If one term has a “-” in front of it, do not forget to transfer it to the product.
Reduce all monomials to standard form. That is, rearrange the factors inside and simplify. For example, the expression 2x*(3.5x) will be equal to (2*3.5)*x*x=7x^2.
Once all the monomials are standardized, try simplifying the polynomial. To do this, group terms that have the same part with variables, for example, (2x+5x-6x)+(1-2). Simplifying the expression, you get x-1.
Pay attention to the presence of parameters in the expression. Sometimes it is necessary to simplify a polynomial as if the parameter were a number.
To convert an expression containing a root into a polynomial, print below it the expression that will be squared. For example, use the formula a^2+2ab+b^2 =(a+b)^2, then remove the root sign along with the even power. If you can't get rid of the root sign, you won't be able to convert the expression to a standard polynomial.
Instructions
Expand all parentheses of the expression. To do this, use formulas, for example, (a+b)^2=a^2+2ab+b^2. If you do not know the formulas, or they are difficult to apply to a given expression, open the brackets sequentially. To do this, multiply the first term of the first expression by each term of the second expression, then the second term of the first expression by each term of the second, etc. As a result, all elements of both brackets will be multiplied together.
If you have three expressions in parentheses, first multiply the first two, leaving the third expression untouched. After simplifying the result obtained by transforming the first parentheses, multiply it with the third expression.
Carefully follow the signs in front of monomial factors. If you multiply two terms with the same sign (for example, both are positive or both are negative), the monomial will have a "+" sign. If one term has a “-” in front of it, do not forget to transfer it to the product.
Reduce all monomials to standard form. That is, rearrange the factors inside and simplify. For example, the expression 2x*(3.5x) will be equal to (2*3.5)*x*x=7x^2.
Once all the monomials are standardized, try simplifying the polynomial. To do this, group terms that have the same part with variables, for example, (2x+5x-6x)+(1-2). Simplifying the expression, you get x-1.
To convert an expression containing a root into a polynomial, print below it the expression that will be squared. For example, use the formula a^2+2ab+b^2 =(a+b)^2, then remove the root sign along with the even power. If you can't get rid of the root sign, you won't be able to convert the expression to a standard polynomial.
Sources:
- polynomial conversion calculator
Brevity, as they say, is the sister of talent. Everyone wants to show off their talent, but his sister is a complicated thing. For some reason, brilliant thoughts take on the form of complex sentences with many adverbial phrases. However, it is up to you to simplify your sentences and make them understandable and accessible to everyone.
Instructions
To make it easier for the recipient (whether listener or reader), try to replace participles and participial phrases short subordinate clauses, especially if there are too many of the above phrases in one sentence. “A cat who came home, having just eaten a mouse, purred loudly, caressed his owner, trying to look into his eyes, hoping to beg for fish brought from the store” - this will not work. Break such a structure into several parts, take your time and don’t try to say everything in one sentence, you’ll be happy.
If you are planning brilliant statement, but it turned out to be too much subordinate clauses(especially with one), then it is better to break the statement into several separate sentences or omit some element. “We decided that he would tell Marina Vasilievna, that Katya would tell Vita that...” - we can continue endlessly. Stop in time and remember who will read or listen to this.
Label different similar members differently. To do this, it is better to emphasize with single, double and triple lines, use color and other line shapes.
Having found all similar members, start combining them. To do this, remove similar terms from the found ones out of brackets. Don't forget that in standard form A polynomial has no such terms.
Check to see if you have any duplicate elements in your entry. In some cases, you may have similar members again. Repeat the operation combining them.
Make sure that the second condition required to write a polynomial in standard form is met: each member must be represented as a monomial in standard form: in the first place is a numerical factor, in the second place is a variable or variables, following in the already indicated order. In this case, it has a letter sequence specified by the alphabet. Decreasing degrees are taken into account secondarily. Thus, the standard form of a monomial is the notation 7xy2, while y27x, x7y2, y2x7, 7y2x, xy27 are not required.
Video on the topic
Mathematical Science studies various structures, sequences of numbers, relationships between them, drawing up equations and solving them. This formal language, which can clearly describe those close to ideal properties real objects studied in other fields of science. One such structure is a polynomial.
Instructions
Polynomial or (from the Greek “poly” - many and the Latin “nomen” - name) – elementary functions classical algebra and algebraic geometry. This is a function of one variable, which has the form F(x) = c_0 + c_1*x + ... + c_n*x^n, where c_i are fixed coefficients, x is a variable.
Polynomials are used in many areas, including the study of zero, negative and complex numbers, the theory of groups, rings, knots, sets, etc. Using polynomial calculations greatly simplifies the expression of properties of different objects.
Basic definitions:
Each term of a polynomial is called a monomial.
A polynomial consisting of two monomials is called a binomial or binomial.
Polynomial coefficients – real or complex numbers.
If the coefficient is equal to 1, then it is called unitary (reduced).
The powers of the variable in each monomial are integers non-negative numbers, maximum degree determines the degree of a polynomial, and its full degree is called an integer, equal to the sum all degrees.
Monomial corresponding zero degree, is called a free member.
A polynomial all of which have the same full degree, is called homogeneous.
Some commonly used polynomials are named after the scientist who defined them, as well as the functions they define. For example, Newton's binomial is for decomposing a polynomial into individual terms to calculate powers. These are the famous ones school curriculum writing the squares of the sum and difference (a + b)^2 – a^2 + 2*a*b + b^2, (a – b)^2 = a^2 – 2*a*b + b^2 and difference squares (a^2 – b^2) = (a - b)*(a + b).
If we allow a polynomial to be written negative powers, then you get a polynomial or Laurent series; The Chebyshev polynomial is used in approximation theory; Hermite polynomial - in probability theory; Lagrange - for numerical integration and interpolation; Taylor - when approximating a function, etc.
note
Newton's binomial is often mentioned in books ("The Master and Margarita") and films ("Stalker"), when the characters decide math problems. This term is well-known and therefore considered the most famous polynomial.
Transformation of expressions is most often done in order to simplify them. For this purpose, special relations are used, as well as rules for reduction and reduction of similar ones.
You will need
- - operations with fractions;
- - abbreviated multiplication formulas;
- - calculator.
Instructions
The simplest transformation is to bring similar ones. If there are terms that are monomials with identical factors, the coefficient for them can be added, taking into account the signs that appear in front of these coefficients. For example, expression 2 n-4n+6n-n=3 n.
If identical factors have degrees, In a similar way it is impossible to combine the likes. Group only those coefficients that have factors with . For example, simplify expression 4 k?-6 k+5 k?-5 k?+k-2 k?=3 k?-k?-5 k.
If possible, use abbreviated multiplication formulas. The most popular are the cube and square of the sum or difference of two numbers. They represent special case Newton. To the formulas for abbreviated multiplication also the squares of two numbers. For example, to find 625-1150+529=(25-23)?=4. Or 1296-576=(36+24) (36-24)=720.
The purpose of the lesson: systematize the knowledge and skills of students to apply the formulas of the squared difference, sum and difference of squares to transform polynomials.
Lesson objectives:
- general education: practicing skills and abilities to transform polynomials using abbreviated multiplication formulas by solving written and oral exercises;
- developing: develop cognitive interest, continue formation mathematical speech, develop the ability to analyze and compare;
- educational: develop the ability to listen to others and the ability to communicate.
Motivational task: create a situation of success in the lesson through praise, stimulation of weak and strong answers.
Organizational forms of communication: collective, group, individual.
During the classes
1st stage. Organizing time.
2nd stage. Motivational conversation with students followed by setting a goal and topic.
Teacher: Guys, we have devoted the last few lessons to studying three abbreviated multiplication formulas. What are these formulas?
We have four more formulas ahead.
But today I suggest you work with these formulas and once again find out how well you understand this topic.
And I would like to start my work with the lines of the wise Confucius:
Three paths lead to knowledge:
The path of reflection is the noblest path,
The path of imitation is the easiest and
The path of experience is the most bitter path.
Think and decide for yourself, guys, which path you will take today in class - it will be your personal choice.
3rd stage. Updating basic knowledge.
Teacher: To make the work more successful, let's remember and repeat the formulas for the square of a sum, the difference of two numbers, and the difference of squares.
I will ask two students to come to the board.
I will ask two students to come to the board.
Assignment to the first student: prove the equality of Diophantus
(a + b)(c + d) = (ac + ab)+(bc – ad).
Assignment to the second student: create a support table (magnetic board).
Collect three formulas from separate fragments:
(a + b) 2 = a + 2ab + b
(a – b) 2 = a – 2ab + b
a 2 – b 2 = (a – b)(a + b)
Frontal work with students.
Teacher: And we, guys, at this time, let's repeat the rules of addition and subtraction rational numbers, because we will need this later in the lesson.
Card:
-/10+5/ -5;
-/(-a +b)/ + b;
-/20*3/: (-12).
Teacher: Guys, let's check the formulas on the magnetic board.
Now, using these formulas, complete the following tasks orally.
Replace * with monomials so that the resulting equality is the identity:
- (* + b) 2 = 4c 2 + * + b 2 ;
- (k – *) 2 = * – * + c 2 ;
- (* + 7c) (7c – *) = 49c 2 – 81a 2
- Calculate:
106 2 – 6 2
71 2 – 61 2 - And in next task you need to check whether the full square is selected correctly:
a 2 + 2a + 2 = (a + 1) 2 + 2
Teacher: Guys, let's go back to the proof of Diophantus's equality and check it.
I suggest you write down this equality in your notebook and check it for the first four consecutive numbers _(1.2.3.4).
4th stage. Work on the topic of the lesson.
Teacher: Guys, what did the student use to prove the equality of Diophantus?
Where else are abbreviated multiplication formulas used?
Let's solve the next problem at the board.
The side of the square is equal to a cm. The length of the rectangle is 2 cm greater than the side of the square, and the width is 2 cm smaller side square. Find the area of the rectangle and compare it with the area of the square.
5th stage. Physical education minute.
6th stage. Work in groups “Star Map”.
Teacher: So, guys, since today we mentioned Diophantus (proved his equality), remember what he did mainly? (Equations).
Fine! I suggest now that you also solve 5 equations in groups, in which you can apply abbreviated multiplication formulas, and also educate yourself in the field of astronomy, that is, find out what the constellations Cepheus and Cassiopeia look like.
Listen to the task.
Here, guys, is a fragment of a star map. Solve the equations and connect in series the stars that correspond to the answers you find.
The work is carried out in groups, so mutual assistance and mutual control are possible.
Cards on the table. Next to each equation is the difficulty level (1, 2, 3, 4). Each of us chooses our level, solves the equation and writes the answer on the card.
Then the constellation is drawn.
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Sample check.
7th stage. Reserve (test)
Classify these polynomials according to the method of factoring them.
Option 1.
EXERCISE. Connect polynomials with their corresponding methods of factorization with lines.
Peer review.
8th stage. Lesson summary.
Teacher: Guys, you have worked quite fruitfully today. Thank you.
But I wanted you to once again, remembering the stages of our lesson, answer my question: where did you apply the abbreviated multiplication formulas, in which case did your work become much easier?
You have 4 more formulas ahead. But that will come later, but now get your homework (numbers from the textbook).
And in conclusion, return to our epigraph. Tell me, which path was more successful for you?
Of course, the path of experience, trial and error is the most hard way, but also the most faithful and worthy.
Therefore, I wish you to go with dignity and receive only good and excellent grades.
Lesson grades.
Important a, b, …, z/
Examples of simplified expressions
- 2*a -7*a
- exp(-7*a)/exp(2*a)
- 1/x + 1/y
- sin(x)^2 + cos(x)^2
Rules for entering functions
In function f Real numbers enter as 7.5 , Not 7,5 2*x- multiplication 3/x- division x^3- exponentiation x+7- addition x - 6— subtraction Function f absolute(x) x(module x or |x|) arccos(x) Function - arc cosine of xarccosh(x) xarcsin(x) Function - arcsine of xarcsinh(x) xarctan(x) Function - arctangent of xarctanh(x) xe Function - e exp(x) Function - exponent of x(same as e^x) floor(x) Function - rounding x log(x) or ln(x) x(To obtain log7(x) log10(x)=log(x)/log(10)) pi sign(x) Function - Sign xsin(x) Function - Sine of xcos(x) Function - Cosine of xsinh(x) xcosh(x) xsqrt(x) Function - Root of xx^2 Function - Square xtan(x) Function - Tangent from xtanh(x) x
Solving Polynomial Equations
The use of equations is widespread in our lives. They are used in many calculations, construction of structures and even sports. Man used equations in ancient times, and since then their use has only increased. The polynomial is algebraic sum products of numbers, variables and their powers. Converting polynomials typically involves two types of problems. The expression needs to be either simplified or factorized, i.e. represent it as the product of two or more polynomials or a monomial and a polynomial.
Also read our article "Solve quadratic equation online"
To simplify the polynomial, give similar terms. Example. Simplify the expression \ Find monomials with the same letter part. Fold them up. Write down the resulting expression: \ You have simplified the polynomial.
For problems that require factoring a polynomial, determine common multiplier given expression. To do this, first remove from brackets those variables that are included in all members of the expression. Moreover, these variables should have the lowest indicator. Then calculate the largest common divisor each of the coefficients of the polynomial. The modulus of the resulting number will be the coefficient of the common multiplier.
Solving math problems online
Factor the polynomial \ Take it out of brackets \ because the variable m is included in each term of this expression and its smallest exponent is two. Calculate the common multiplier factor. It is equal to five. Thus, the common factor of this expression is \ Hence: \
Where can I solve a polynomial equation online?
You can solve the equation on our website pocketteacher.ru. Free online solver will allow you to solve online equations of any complexity in a matter of seconds. All you need to do is simply enter your data into the solver. You can also watch video instructions and learn how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group: pocketteacher. Join our group, we are always happy to help you.
Converting Expressions. Briefly about the main thing.
Simplifying Expressions
Step 1: Enter an expression to simplify
The service (a kind of program for grades 5 and 7, 8, 9, 10, 11) allows you to simplify mathematical expressions: algebra ( algebraic expressions), trigonometric expressions, expressions with roots and other powers, reducing fractions, also simplifies complex literal expressions,
to simplify complex expressions that way(!)
Important In expressions, variables are designated by ONE letter! For example, a, b, …, z/
Examples of simplified expressions
- 2*a -7*a
- exp(-7*a)/exp(2*a)
- 1/x + 1/y
- sin(x)^2 + cos(x)^2
Rules for entering functions
In function f you can do the following operations: Real numbers enter as 7.5 , Not 7,5 2*x- multiplication 3/x- division x^3- exponentiation x+7- addition x - 6— subtraction Function f may consist of functions (designations are given in alphabetical order): absolute(x) Function - absolute value x(module x or |x|) arccos(x) Function - arc cosine of xarccosh(x) Function - hyperbolic arc cosine of xarcsin(x) Function - arcsine of xarcsinh(x) Function is the hyperbolic arcsine of xarctan(x) Function - arctangent of xarctanh(x) Function is the hyperbolic arctangent of xe Function - e this is the one that is approximately equal to 2.7 exp(x) Function - exponent of x(same as e^x) floor(x) Function - rounding x downward (example floor(4.5)==4.0) log(x) or ln(x) Function - Natural logarithm from x(To obtain log7(x), you need to enter log(x)/log(7) (or, for example, for log10(x)=log(x)/log(10)) pi The number is "Pi", which is approximately equal to 3.14 sign(x) Function - Sign xsin(x) Function - Sine of xcos(x) Function - Cosine of xsinh(x) Function - Hyperbolic sine of xcosh(x) Function — Hyperbolic cosine of xsqrt(x) Function - Root of xx^2 Function - Square xtan(x) Function - Tangent from xtanh(x) Function — Tangent hyperbolic from x
To main
School algebra
Polynomials
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, a monomial is a special case of a polynomial; it consists of one term.
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 a standard form; among the monomials there are no similar ones.
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 have given polynomial standard type.
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 5a2h3s4 +1? The degree of the polynomial 5a2h3s4 + 1 is equal to nine, because this polynomial contains two monomials, greatest degree has the first monomial 5a2h3s4, and its degree is 9.
Solving Polynomial Equations
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.