Solve some complicated problemsFactorizationThe problem is often used in the substitution method, that is, if some parts of a polynomial with complex structure are regarded as a whole and replaced by new letters (that is, substitution), the complex problem can be simplified, clarified and reducedpolynomialThe number of terms can reduce the complexity of polynomial structure[1]。
The substitution method, also known as variable substitution method, is ourProblem solvingOne of the commonly used methods.Using the substitution method, you cansimplify what is complicatedTo find a shortcut to solve the problem.
Also called auxiliary unknown method, also called variable substitution method. It is an important method for solving equations.It is a widely used method, and its general meaning is to use new variables to express a part of the mathematical expression composed of one or several variables, so as to facilitate the solution of the problem. Here, only the application in solving equations (groups) and solving inequalities (groups) is given[2]。
The substitution method is to introduce one or several new variables to replace the original variables of some variables to obtain the results, and then return to the original variables to obtain the results. By introducing new elements, the substitution method connects the scattered conditions, or displays the implicit conditions, or links the conditions with the conclusions, or becomes a familiar problem. Its theoretical basis is equivalent substitution
There are two main types of substitution methods in high school mathematics:
(1) Overall exchange of yuan: exchange "yuan" for "formula".
(2) Triangle for yuan, "formula" for "yuan".
(3) In addition, there are symmetrical substitution, mean substitution, universal substitution, etc. The substitution method is widely used.For example, solving equations, solving inequalities, proving inequalities, finding the range of functions, finding the general term and sum of number sequences, etc. In addition, it is also widely used in analytic geometry.
Application Skills
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When we use the substitution method, we should follow the principle ofStandardizationThe selection of new variable range shall be noted after changing the elementValue rangeCorresponds toprimaryThe range of variable values cannot be narrowed or reducedexpand。T>0 and sin α ∈ [- 1,1] in the above examples.
You can observe firstArithmeticIt can be found that the formula requiring substitution always contains the sameFormulaAnd then use themOneletterreplace, deduce the answer, and then if there is this letter in the answer, bring the formula into it,SoCan be calculated[3]。
Factorization
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Sometimes inFactorizationYou can selectpolynomialThe same part in is replaced by another unknown number, then factorized, and finally converted back. This method is called the substitution method.
Related examples
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Example 1
Note: Do not forget to return the yuan after changing it.
[Example] When decomposing (x ²+x+1) (x ²+x+2) - 12, you can make y=x ²+x, then the original formula=(y+1) (y+2) - 12=y ²+3y+2-12=y ²+3y-10=(y+5) (y-2)=(x ²+x+5) (x ²+x-2)=(x ²+x+5) (x+2) (x+2) (x+2) (x-1)
Sometimes insolve equationsYou can select the same part of the equation to replace it with another unknown number, so as to reduce the order, then carry out the new equation to find the new unknown number, and finally convert it back to the original unknown number. This method is called the substitution method.
Example 2
Note: Do not forget to return the yuan after changing it.
