In fact, the classification of algebraic expressions can be divided into two categories: fractions and algebraic expressions. Fractions are algebraic expressions with letters in the denominator, so algebraic expressions are algebraic expressions without letters in the denominator (of course, algebraic expressions do not necessarily contain fractions. )
Algebraic expressions can be divided into monomials and polynomials. The monomial refers to the algebraic expression with multiplication and division, and the polynomial refers to the algebraic expression with addition and subtraction besides maturity. Of course, polynomials can also be understood as the addition or subtraction of several monomials.
So what we began to explore was the addition and subtraction of algebraic expressions and algebraic expressions. In fact, the essence is to find similar items, which can be added or subtracted. What are similar items? Similar terms refer to terms with the same base and exponent, or terms with the same letters but exponent, that is, similar terms do not always appear in the form of power. Similar terms can also refer to monomials. For example, ab and ab are similar terms, and ab is obviously a monomial.
After reviewing the addition and subtraction of algebraic expressions, the definition of similar terms and the classification of algebraic expressions, we will classify the multiplication and division of algebraic expressions.
When discussing algebraic multiplication, I think it can be divided into three sections. 1 is a monomial multiplication, the second is a monomial multiplication polynomial, and the third is a polynomial multiplication polynomial.
When discussing algebraic division, I think it can be divided into four sections. 1 is a single division, the second section is a polynomial division, the third section is a single division polynomial, and the fourth section is a polynomial division polynomial.
The first thing we want to discuss is multiplication.
Then there are two types of 1 problem that we encounter, that is, the problem of monomial multiplying monomial, which can be simplified and cannot be simplified.
Can not be simplified, for example: ab×cd, obviously the final answer is abcd, in fact, this calculation is completely different from not calculating, just omitting a symbol.
What can be simplified after calculation results can be roughly divided into three categories. That is, the power of the same base, the power of the product.
Multiplication with the same base is like: a? ×a？ , the following is the calculation process:
Answer? ×a？
=(aa)(aaaaa)
=aaaaaa
=a？
Another example: b? ×b？ , the following is the calculation process:
b？ ×b？
=(bbbb)(bbbbbbbbb)
=bbbbbbbbbbbbb
=b
Through observation, it can be found that the exponent of the result is actually the sum of the exponents of two multipliers, which shows that same base powers's multiplication rule is: the base is constant, and the exponents are added. So is it proved by rigorous mathematical reasoning? Such as: a? ×a？
Reasoning process:
Answer? ×a？
=aaa…( n a)×aaa…( y A)
= aaaaa...(n+y ace。 )
=a
Prove completion.
Then there is motivation, such as: (a? )? , the following is the calculation process:
(a？ )?
=a？ Answer? Answer? Answer?
=a (according to the multiplication rule of the same radix power)
Another example is: (a? )?
=a？ Answer?
=a (according to the multiplication rule of the same radix power)
It can be seen that the general form of power is: (a? )? When calculating the previous special case, we will find that the index of the result is the product of two multiplier indexes, so we can judge.
(a？ )? =a
So how to prove it by mathematical logic reasoning?
(a？ )?
=a？ Answer? Answer? Answer? Answer? ...…(y ace？ Multiply)
=a (according to the law of power multiplication with the same base)
Then there is the power of the product, such as: (ab)? , the following is the calculation process:
(ab)？
=(ab)(ab)(ab)
=aaabbb (by multiplicative commutative law)
=a？ b？ (According to the law of the same base power multiplication)
Therefore, we can summarize the general form of the power of the product as follows: (ab)? According to experience, we can infer that the result is: a? Answer? Because by observing special cases, we will find that the power of the product of two numbers is equal to the product of the power of two numbers. So how should we prove it with rigorous mathematical reasoning?
(ab)？
= Baba Baba ... (n ab times)
= aaaaa ... bbbb ... (n) ... (nb times)
=a？ Answer?
In fact, we will find that the power of power and the power of products have the same part. In fact, they all evolved from the form of power.
The form of normal power supply is as follows: a?
The power of the power is actually to replace the radix here with a power, for example, to replace A with the Y power of A, then its form becomes:
(a？ )?
The power of the product actually turns the base of the power form into a product, for example, turning A into ab. Then its form becomes:
(ab)？
So what if we change the radix to the addition formula? This is actually related to polynomial multiplication, which I will discuss later.
In fact, you only need to know the operation steps of polynomial multiplication monomial. First of all, we only need to give a general form: ab(ab+na? B) In step 1, we will use the multiplication and distribution law to open the brackets, which will change this formula into
abab+na？ Babu (female name)
Then through the multiplication rule of the same base power, we can convert this formula into:
Answer? b？ +na？ b？
Therefore, the general steps of multiplying a polynomial by a monomial are:
1. Decompose brackets by multiplication and distribution.
2. Simplify with the multiplication rule of the same base number, and finally get the final result.
Finally, polynomial multiplication can be discussed first, that is, at the beginning, one-way one-way multiplication is what we are involved in, turning the base of a power into an addition formula, such as:
(a+b)？
If a+b is regarded as a whole, it is a power. However, if we regard a+B as a polynomial, then this formula is a polynomial multiplication polynomial.
So how to calculate this formula?
We can still convert it into the form of multiplication of two numbers.
(a+b)(a+b)
Next, we use the multiplication and division method to regard the second a+B as a whole, which will become:
a(a+b)+b(a+b)
If we use the law of multiplication and distribution again, it will become:
aab+ab+b？
By using the law of addition of algebraic expressions and combining similar terms, it will become:
a2ab+b？
So, (a+b)? =a2ab+b？
This is actually a very common formula, because many polynomials can be multiplied by polynomials to become this form, so this formula is called complete sum of squares.
Then we can contact the power supply. The last contact is to replace the radix of the power with the addition formula. What if it's a subtraction formula? For example: (a-b)?
First of all, we still need to convert it into a multiplication formula, that is:
(A-B) A-B
We take the second a-b as a whole, and then remove the brackets by multiplication and division, and this formula becomes:
A (A-B)-B (A-B)
Open the parentheses again using the multiplicative distribution rate:
Answer? -ab-ab+b？
Then use the algebraic subtraction rule, it becomes:
Answer? -2ab+b？
This is actually a formula, because there are many polynomials multiplied by polynomials that can be converted into this form. This polynomial is called complete square difference.
Of course, there is a more magical polynomial multiplication polynomial, similar to: (a-b)(a+b)
First, consider a+b as a whole, and then remove the 1 bracket by multiplication and division:
a(a+b)-b(a+b)
Then by multiplication, the distribution rate becomes:
Answer? +ab-ab-b？
The addition of ab and the subtraction of ab cancel each other out, so the formula becomes:
Answer? -B?
This formula is called the square difference formula.
Next, we will discuss the division of algebraic expressions.
First of all, we use monomials to divide monomials. First of all, we still have to summarize a calculation process. We can still give a practical example, such as 3a. b？ Ab, we can turn it into a fraction and then restore it. We can change this division formula to: 3a? b？ /ab First, we divide the numerator and denominator by ab at the same time, and this score will become: 3ab? /1, then, 3a? b？ ÷ab=3ab？ .
What about dividing by ab and dividing by 3ab? In fact, the numerator and denominator are divided by a 3 at the same time, so, 3a? b？ ÷3ab=ab？
From this, we can sum up the rule that when the coefficient is divided, the base of same base powers degree remains unchanged.
This is a general step to solve the problem of dividing the monomial by the monomial.
Next is the polynomial divided by the monomial, for example:
When doing these questions, in fact, the first thing we have to do is to extract the factor, that is, to convert the dividend polynomial into a product, which will help us to lower the score.
For example, when doing the 1 problem, we can put forward the factor m, and then the formula becomes: m(5m? n？ 6 million? ) ÷3m, then, m is cancelled, so this formula becomes:
(5m？ n？ 6 million? )÷3
Then we just need to remove the brackets according to the law of multiplication and distribution, and then calculate it. Then it became very simple. As for the other topics I just shot, I can do the same.
Let's not discuss the problem of polynomial divided by monomial for the time being.
Finally, the problem to be discussed is polynomial divided by polynomial. We can give some special examples:
We will find that the first three questions are actually related to complete sum of squares, complete difference of squares and mean variance. For example, the title 1 can be changed to: (a+b)? , then, (a+b)? (a+B) is also equal to a+B.
Question 2: Dividends can be changed to: (3a-b)?
The third question can turn the dividend into:
(4m-3n)(4m+3n)
What about the fourth question? This problem is more troublesome. First of all, we still have to try to change the dividend into multiplication form. We will find that dividends are paid by a? And 1, first we can put one? Become different from the divisor a(a? +a+ 1) here, we will put a? One more a? And a, so we have to subtract a So we'll be a? become
a(a？ +a+ 1)-a？ [Ancient names or Latin modern names of animals and plants]
Because dividends are not just. Therefore, the dividend as a whole should become: a(a? +a+ 1)-a？ -a- 1, we will find that this dividend has a * * * same factor, that is:
Answer? -a- 1
So we can turn dividends into:
(a- 1)(a？ +a+ 1)
Then the formula of the original problem becomes:
(a- 1)(a？ +a+ 1)÷(a？ +a+ 1, so the final answer is a- 1.
So, if the multiplication and division method is reciprocal, the formula becomes: a? - 1=(a- 1)(a？ +a+ 1), do you feel that this formula is very similar to the previous formula? Yes, with a? -B? The square is very similar, because 1 is actually equal to the cube of 1, so this formula can also be expressed like this:
(a- 1)？ =(a- 1)(a？ +a+ 1) This is actually a special case of cubic difference formula. The general form of cubic difference formula should be:
(a-b)?
So what should this formula become after the speech? It will actually become: a? -3a？ b+3ab？ -B?
So what should be the general form of the cubic sum formula? It should be: (a+b)? After removing the brackets, it becomes: a? +3a？ b+3ab？ +b？
What we will study in the future is actually a more detailed study of factorization and division of algebraic expressions. What is a factorization factor? If we use the square difference formula to explain it, that is, from a? -B? The shape of the square is converted into
The form of (a-b)(a+b).
Then, will algebraic multiplication and division be used in some practical problems? The most common problem is to find the area, such as this question:
This question is actually testing the square difference formula we have learned.
This is about multiplication and division of algebraic expressions.