With this math program, you can divide polynomials with a column.
The program for dividing a polynomial by a polynomial does not just give the answer to the problem, it gives a detailed solution with explanations, i.e. displays the solution process in order to check the knowledge of mathematics and / or algebra.
This program can be useful for high school students general education schools in preparation for control works and exams, when checking knowledge before the exam, parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to do as quickly as possible homework in math or algebra? In this case, you can also use our programs with a detailed solution.
In this way, you can conduct your own training and / or the training of your younger brothers or sisters, while the level of education in the field of the problems being solved rises.
If you need or simplify polynomial or multiply polynomials, then for this we have a separate program Simplification (multiplication) of the polynomial
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A bit of theory.
Division of a polynomial by a polynomial (binomial) by a column (angle)
In algebra division of polynomials by a column (corner)- an algorithm for dividing the polynomial f (x) by a polynomial (binomial) g (x), the degree of which is less than or equal to the degree of the polynomial f (x).
The algorithm for dividing a polynomial by a polynomial is a generalized form of dividing numbers by a column, easily implemented by hand.
For any polynomials \ (f (x) \) and \ (g (x) \), \ (g (x) \ neq 0 \), there are unique polynomials \ (q (x) \) and \ (r (x ) \) such that
\ (\ frac (f (x)) (g (x)) = q (x) + \ frac (r (x)) (g (x)) \)
and \ (r (x) \) has more low degree than \ (g (x) \).
The goal of the algorithm for dividing polynomials into a column (angle) is to find the quotient \ (q (x) \) and remainder \ (r (x) \) for a given dividend \ (f (x) \) and nonzero divisor \ (g (x) \)
Example
We divide one polynomial by another polynomial (binomial) by a column (corner):
\ (\ large \ frac (x ^ 3-12x ^ 2-42) (x-3) \)
The quotient and remainder of the given polynomials can be found by performing the following steps:
1. Divide the first element of the dividend by the leading element of the divisor, place the result under the line \ ((x ^ 3 / x = x ^ 2) \)
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3. Subtract the polynomial obtained after multiplication from the dividend, write the result under the line \ ((x ^ 3-12x ^ 2 + 0x-42- (x ^ 3-3x ^ 2) = - 9x ^ 2 + 0x-42) \)
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4. We repeat the previous 3 steps, using the polynomial written under the line as the dividend.
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5. Repeat step 4.
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6. End of the algorithm.
Thus, the polynomial \ (q (x) = x ^ 2-9x-27 \) is the quotient of the division of polynomials, and \ (r (x) = - 123 \) is the remainder of the division of polynomials.
The result of dividing polynomials can be written as two equalities:
\ (x ^ 3-12x ^ 2-42 = (x-3) (x ^ 2-9x-27) -123 \)
or
\ (\ large (\ frac (x ^ 3-12x ^ 2-42) (x-3)) = x ^ 2-9x-27 + \ large (\ frac (-123) (x-3)) \)
In school, these actions are studied from simple to complex. Therefore, it is imperative that you learn well the algorithm for performing these operations on simple examples... So that later there are no difficulties with division decimal fractions in a column. After all, this is the most difficult version of such tasks.
This subject requires consistent study. Knowledge gaps are unacceptable here. This principle should be learned by every student already in the first grade. Therefore, if you skip several lessons in a row, you will have to master the material yourself. Otherwise, later there will be problems not only with mathematics, but also with other subjects related to it.
The second prerequisite for successful study of mathematics is to switch to long division examples only after you have mastered addition, subtraction and multiplication.
It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to learn it according to the Pythagorean table. There is nothing superfluous, and multiplication is assimilated in this case easier.
How are natural numbers multiplied in a column?
If there is a difficulty in solving examples in a column for division and multiplication, then you should start to fix the problem with multiplication. Since division is the inverse of multiplication:
- Before you multiply two numbers, you need to look at them carefully. Choose the one with more digits (longer), write it down first. Place the second under it. Moreover, the numbers of the corresponding category should be under the same category. That is, the rightmost digit of the first number should be above the rightmost digit of the second.
- Multiply the rightmost digit bottom number for each digit on the top, starting from the right. Write the answer under the line so that its last digit is under the one multiplied by.
- Repeat the same with the other digit of the lower number. But the result from the multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.
Continue this multiplication in a column until the numbers in the second multiplier run out. Now they need to be folded. This will be the desired answer.
Algorithm for multiplication in a column of decimal fractions
First, it is supposed to imagine that not decimal fractions are given, but natural ones. That is, remove commas from them and then proceed as described in the previous case.
The difference begins when the answer is recorded. At this moment, it is necessary to count all the numbers that come after the commas in both fractions. That is how many of them you need to count from the end of the answer and put a comma there.
It is convenient to illustrate this algorithm with an example: 0.25 x 0.33:
Where to start learning division?
Before solving the long division examples, it is necessary to remember the names of the numbers that stand in the division example. The first of these (the one that is divided) is the dividend. The second (divided by) is the divisor. The answer is private.
After that, on simple everyday example let's explain the essence of this mathematical operation... For example, if you take 10 candies, then it is easy to divide them equally between mom and dad. But what if you need to distribute them to parents and brother?
After that, you can get acquainted with the division rules and master them on specific examples... First, simple, and then move on to more and more complex.
Algorithm for dividing numbers into a column
First, we present the procedure for natural numbers divisible by a single digit. They will also be the basis for multi-digit divisors or decimal fractions. Only then is it supposed to make small changes, but more on that later:
- Before doing long division, you need to figure out where the dividend and the divisor are.
- Write down the dividend. To the right of it is the divider.
- Draw a corner to the left and below near the last.
- Determine the incomplete dividend, that is, the number that will be the minimum for division. It usually consists of one digit, maximum two.
- Choose the number that will be the first to be written in the answer. It should be the number of times the divisor fits in the dividend.
- Write the result from multiplying this number by the divisor.
- Write it under an incomplete dividend. Subtract.
- Remove to the remainder the first digit after the part that has already been divided.
- Pick up the number for the answer again.
- Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: demolish a digit, pick up a number, multiply, subtract.
How to solve long division if there is more than one digit in the divisor?
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then it is supposed to work with the first three digits.
There is one more nuance in this division. The fact is that the remainder and the digit taken down to it are sometimes not divisible by the divisor. Then it is supposed to assign one more figure in order. But at the same time, you must put zero in the answer. If you are dividing three-digit numbers into a column, then it may be necessary to demolish more than two digits. Then a rule is introduced: there should be one less zeros in the answer than the number of digits removed.
You can consider such a division using the example - 12082: 863.
- The incomplete divisible in it turns out to be the number 1208. The number 863 is placed in it only once. Therefore, in response, it is supposed to put 1, and under 1208, write 863.
- Subtraction gives a remainder of 345.
- To him you need to demolish the number 2.
- Of the 3452, 863 fits four times.
- A four must be written in response. Moreover, when multiplied by 4, this is the number obtained.
- The remainder after subtraction is zero. That is, the division is over.
The answer in the example will be the number 14.
What if the dividend ends in zero?
Or a few zeros? In this case, a zero remainder is obtained, and there are still zeros in the dividend. You should not despair, everything is easier than it might seem. It is enough to simply assign all the zeros that were not separated to the answer.
For example, you need to divide 400 by 5. Incomplete dividend 40. Five is placed in it 8 times. This means that the answer is supposed to write 8. When subtracting the remainder, there is no remainder. That is, the division is complete, but zero remains in the dividend. It will have to be attributed to the answer. So when you divide 400 by 5, you get 80.
What if you need a decimal to divide?
Again, this number looks like a natural number, if not for the comma separating the integer part from the fractional part. This suggests that long divisions are similar to the one described above.
The only difference is the semicolon. It is supposed to be answered as soon as the first digit from the fractional part is taken down. In another way, it can be said this way: the division of the whole part has ended - put a comma and continue the solution further.
When solving examples for long division with decimal fractions, you need to remember that in the part after the decimal point, you can assign any number of zeros. Sometimes this is needed in order to complete the numbers to the end.
Division of two decimal fractions
It may sound complicated. But only at the beginning. After all, how to perform column division of fractions by natural number, is already clear. Hence, it is necessary to reduce this example to the already familiar form.
This is easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million, if the task requires it. The factor is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, as a result, it turns out that the fraction will have to be divided by a natural number.
And this will be the worst case. After all, it may happen that the dividend from this operation becomes an integer. Then the solution of the example with column division of fractions will be reduced to the very simple option: operations with natural numbers.
As an example, divide 28.4 by 3.2:
- First, they must be multiplied by 10, since there is only one digit in the second number after the decimal point. Multiplication will give 284 and 32.
- They are supposed to be separated. Moreover, the whole number is 284 by 32 at once.
- The first matched number for the answer is 8. It multiplies 256. The remainder is 28.
- The division of the whole part is over, and in response it is supposed to put a comma.
- Carry out to remainder 0.
- Take 8 again.
- Remainder: 24. Add one more 0 to it.
- Now you need to take 7.
- The result of the multiplication is 224, the remainder is 16.
- Take down another 0. Take 5 each and you get exactly 160. The remainder is 0.
The division is over. The result of example 28.4: 3.2 is 8.875.
What if the divisor is 10, 100, 0.1, or 0.01?
As with multiplication, long division is not needed here. It is enough just to move the comma in the desired direction by a certain number of digits. Moreover, according to this principle, you can solve examples with both whole numbers and decimal fractions.
So, if you need to divide by 10, 100 or 1,000, then the comma is shifted to the left by as many digits as there are zeros in the divisor. That is, when a number is divisible by 100, the comma must move two digits to the left. If the dividend is a natural number, then it is assumed that the comma is at its end.
This action gives the same result as if the number were to be multiplied by 0.1, 0.01, or 0.001. In these examples, the comma is also wrapped to the left by the number of digits equal to the length of the fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the comma must move to the right one digit (or two, three, depending on the number of zeros or the length of the fractional part).
It is worth noting that the number of digits given in the dividend may be insufficient. Then, to the left (in the integer part) or to the right (after the decimal point), you can assign the missing zeros.
Division of periodic fractions
In this case, you will not be able to get an exact answer with long division. How to solve an example if a fraction with a period is encountered? Here we are supposed to go over to ordinary fractions. And then perform their division according to the previously learned rules.
For example, you need to divide 0, (3) by 0.6. The first fraction is periodic. It is converted to 3/9, which, when canceled, will give 1/3. The second fraction is the final decimal. It is even easier to write it down as an ordinary one: 6/10, which is equal to 3/5. The division rule for ordinary fractions prescribes to replace division by multiplication and divisor - by its reciprocal. That is, the example boils down to multiplying 1/3 by 5/3. The answer is 5/9.
If the example has different fractions ...
Then several solutions are possible. At first, common fraction you can try to convert to decimal. Then divide two decimal places according to the above algorithm.
Secondly, each final decimal fraction can be written in the form of an ordinary one. Only it is not always convenient. Most often, these fractions are huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.
Long division is an integral part teaching material junior student. Further success in mathematics will depend on how well he learns to perform this action.
How to properly prepare a child for the perception of new material?
Long division is a complex process that requires a child to certain knowledge... To perform division, you need to know and be able to quickly subtract, add, and multiply. Knowledge of the digits of numbers is also important.
Each of these actions should be brought to automatism. The child should not think for a long time, as well as be able to subtract, add not only the numbers of the first ten, but within a hundred in a few seconds.
It is important to form the correct concept of division as a mathematical action. Even when studying the multiplication and division tables, the child should clearly understand that the dividend is a number that will be divided into equal parts, the divisor is to indicate how many parts the number needs to be divided into, the quotient is the answer itself.
How to explain the algorithm of mathematical actions step by step?
Each mathematical action presupposes strict adherence to a certain algorithm. Long division examples should be performed in this order:
- Writing an example in a corner, while the places of the dividend and divisor must be strictly observed. To help the child not get confused in the early stages, we can say that on the left we write more, and on the right - the smaller one.
- Allocate the part for the first division. It must be divisible with a remainder.
- Using the multiplication table, we determine how many times the divider can fit in the selected part. It is important to point out to the child that the answer should not exceed 9.
- Perform multiplication of the resulting number by the divisor and write it down on the left side of the corner.
- Next, you need to find the difference between the part of the dividend and the resulting product.
- The resulting number is written under the line and demolished the following bit number... Such actions are performed until the period until the remainder is 0.
A clear example for the student and parents
Long division can be clearly explained with this example.
- Write down 2 numbers in a column: the dividend - 536 and the divisor - 4.
- The first part for division must be divisible by 4 and the quotient must be less than 9. The number 5 is suitable for this.
- 4 fits in 5 only 1 time, so in the answer we write 1, and under 5 - 4.
- Further, subtraction is performed: 4 is subtracted from 5 and 1 is written under the line.
- The next digit number is taken down to one - 3. In thirteen (13) - 4 will fit 3 times. 4x3 = 12. Twelve is written under the 13th, and 3 - in the quotient, as the next digit number.
- Subtract 12 from 13, and get 1 in the answer. Again, take down the next digit number - 6.
- 16 is again divisible by 4. In response, write down 4, and in the division column - 16, draw a line and in the difference 0.
Solving long division examples with your child multiple times can help you get things done quickly in high school.
How to divide decimal fractions by natural numbers? Let's consider the rule and its application with examples.
To divide a decimal fraction by a natural number, you need:
1) divide a decimal fraction by a number, ignoring the comma;
2) when the division of the whole part ends, put a comma in the quotient.
Examples.
Split decimals:
To divide a decimal fraction by a natural number, divide without paying attention to the comma. 5 is not divisible by 6, so we put zero in the quotient. The division of the whole part is over, we put a comma in the quotient. We demolish zero. We divide 50 by 6. Take 8.6 ∙ 8 = 48 each. We subtract 48 from 50, and the remainder gets 2. We demolish 4. 24 divide by 6. We get 4. The remainder is zero, which means that the division is over: 5.04: 6 = 0.84.
2) 19,26: 18
Divide the decimal fraction by a natural number, ignoring the comma. Divide 19 by 18. Take 1. The division of the whole part is over, in the quotient we put a comma. Subtract from 19 18. In the remainder - 1. We demolish 2. 12 is not divisible by 18, in the quotient we write zero. We demolish 6. 126 we divide by 18, we get 7. The division is over: 19.26: 18 = 1.07.
Divide 86 by 25. Take 3.25 ∙ 3 = 75. Subtract 75 from 86. The remainder is 11. The division of the whole part is over, in the quotient we put a comma. We demolish 5. Take 4. 25 ∙ 4 = 100. Subtract 100 from 115. The remainder is 15. We demolish zero. We divide 150 by 25. We get 6. The division is over: 86.5: 25 = 3.46.
4) 0,1547: 17
Zero is not divisible by 17, in the quotient we write zero. The division of the whole part is over, we put a comma in the quotient. We demolish 1. 1 by 17 is not divisible, in the quotient we write zero. We demolish 5. 15 by 17 is not divisible, in the quotient we write zero. We demolish 4. Divide 154 by 17. Take 9.17 ∙ 9 = 153. Subtract 153 from 154. In the remainder - 1. Take down 7. Divide 17 by 17. We get 1. The division is over: 0.1547: 17 = 0.0091.
5) The decimal fraction can also be obtained when dividing two natural numbers.
When dividing 17 by 4, we take 4. The division of the whole part is over, in the quotient we put a comma. 4 ∙ 4 = 16. Subtract 16 from 17. The remainder is 1. Take zero. Divide 10 by 4. Take 2.4 ∙ 2 = 8. Subtract 8 from 10. The remainder is 2. We demolish zero. Divide 20 by 4. Take 5. The division is over: 17: 4 = 4.25.
And a couple more examples for dividing decimal fractions by natural numbers:
Instructions
Test your child's multiplication skills first. If a child does not know the multiplication table firmly, then he may also have problems with division. Then, when explaining the division, you can be allowed to pry into the cheat sheet, but you still have to learn the table.
Write the dividend and divisor, separated by the separating vertical bar. Under the divisor, you will write the answer - quotient, separating it with a horizontal line. Take the first digit of 372 and ask your child how many times the number six "fits" in a three. That's right, not at all.
Then take already two numbers - 37. For clarity, you can highlight them with a corner. Again, repeat the question - how many times is the number six contained in 37. It is useful to count quickly. Pick up the answer together: 6 * 4 = 24 - completely different; 6 * 5 = 30 - close to 37. But 37-30 = 7 - six "fit" again. Finally, 6 * 6 = 36, 37-36 = 1 - fits. The first digit of the quotient found is 6. Write it under the divisor.
Write 36 under the number 37, draw a line. For clarity, you can use the sign in the entry. Put the remainder under the line - 1. Now "lower" the next digit of the number, two, to one - it turned out 12. Explain to the child that the numbers always "descend" one at a time. Again ask how many "sixes" there are 12. The answer is 2, this time without a remainder. Write the second digit of the quotient next to the first. The final result is 62.
Also consider the case of division in detail. For example, 167/6 = 27, remainder 5. Most likely, your son hasn't heard anything about simple fractions yet. But if he asks questions, with the remainder further, it can be explained by the example of apples. 167 apples were shared among six people. Each got 27 pieces, and five apples were left unshared. You can divide them too, cutting each into six slices and distributing them equally. Each person got one slice from each apple - 1/6. And since there were five apples, each had five slices - 5/6. That is, the result can be written like this: 27 5/6.
To consolidate the information, analyze three more examples of division:
1) The first digit of the dividend contains the divisor. For example, 693/3 = 231.
2) The dividend ends in zero. For example, 1240/4 = 310.
3) The number contains a zero in the middle. For example, 6808/8 = 851.
In the second case, children sometimes forget to add the last digit of the answer - 0. And in the third, it happens that they jump over zero.
Sources:
- column division grade 3
- How to divide 927 long
Concrete meanings are learned by children much better than abstract ones. How to explain to kid what are two thirds? Concept fractions requires a special introduction. There are some methods to help you understand what a non-integer number is.
You will need
- - special lotto;
- - apple and candy;
- a circle of cardboard, consisting of several parts;
- - crayon.
Instructions
Try to interest. Play some special classics as you walk. If you are tired of jumping into ordinary ones, and the child has mastered counting well, try this option. Draw the classics with chalk on the asphalt as shown in the picture and explain to the kid what to jump like this: 1 - 2 - 3 ... or you can do so 1 - 1.5 - 2 - 2.5 ... Children really like to play and so they are better, that between the numbers, there are also intermediate values - parts. This is your step towards learning fractional numbers. Excellent visual aid.
Take a whole apple and offer it to two at the same time. They will tell you right away that this is impossible. Then cut up the apple and offer them again. Now everything is all right. each got the same half of an apple. These are the parts of one whole.
Offer to split four in half with you. He can do it easily. Then get another one and offer to do the same. It is clear that you cannot get a whole candy right away and to kid... A way out can be found by cutting the candy in half. Then each will have two whole candies and one half.
For older ones, use a cut wheel. It can be divided into 2, 4, 6 or 8 parts. We invite the children to take a circle. Then we divide it into two halves. A circle will turn out great from two halves, even if you exchange a half with a neighbor on a desk (the circles should be of the same diameter). We divide each half of the loan into half. It turns out that the circle can also consist of 4 parts. And each half is obtained from two halves. Then we write it down on the board in the form fractions... Explaining what the numerator (parts were taken) and the denominator (how many parts were all divided) are. So it is easier for children to learn a difficult concept - a fraction.
Be sure to apply visual aids in explaining an abstract concept.
Section "Multiplication and Division" - one of the most difficult in the course of mathematics primary grades... Her children usually study at the age of 8-9 years. At this time, they have a well-developed mechanical memory, so memorization occurs quickly and without much effort.
