Doing Olympic math problems is helpful to improve our ability, not only in mathematics, but also in other aspects, mainly to make us think more. Let's share the common methods of algebraic evaluation of Olympic numbers. Let's have a look!
When calculating the algebraic value, we can directly substitute it into the calculation, or simplify it first and then evaluate it. The latter is often simpler than the former. To calculate the value of algebraic expression according to known conditions, it is necessary to correctly grasp the overall characteristics of algebraic expression and flexibly choose appropriate methods to solve it. Here are some examples.
First, direct substitution evaluation
Example 1 When x=-2 and y= 1, the value of the algebraic expression x2-xy is.
Solution: When x=-2 and y= 1, x2-xy = (-2) 2-(-2)&; Times; 1=6. So this question should be filled in: 6.
Note: When there are no similar items in a given algebraic expression, the values of letters are often directly substituted into it for evaluation.
Second, simplify first, and then substitute for evaluation.
Example 2 calculation: 5m2-[3m-(2m-3)+5m2], where m=-3.
Solution: Method 1: Original formula = 5m2-[3m-2m+3+5m2]
= 5m2-(m+3+5m2)
= 5m2-3m2-5m2
=(5m2-5m2)-m-3
=-m-3。
When m=-3, the original formula = -m-3=3-3=0.
Method 2: The original formula =5m2-3m+(2m-3)-5m2.
=(5m2-5m2)-3m+(2m-3)
=-3m+2m-3
= -m-3。
When m=-3, the original formula = -m-3=3-3=0.
Note: If algebraic expressions can be simplified, it is often easier to evaluate them after simplification. When using the rule of deleting parentheses, you can delete parentheses from the inside out or from the outside in, paying special attention to the sign change when deleting parentheses. In the process of removing brackets, if similar items are encountered, they should be merged first.
Third, apply the whole idea to find the value of algebraic expression.
Example 3 is known: n=- 1. Find the value of algebraic expression 2 (N2-2n+1)-(N2-2n+1)+3 (N2-2n+1).
Analysis: By carefully observing the overall characteristics of a given algebraic expression, it is not difficult to find that every term has n2-2n+ 1. So let's consider merging (n2-2n+ 1) as a whole.
Solution: Original formula =(2- 1+3)(n2-2n+ 1)
=4(n2-2n+ 1)。
When n=- 1, N2-2n+1= (-1) 2-2&; Times; (-1)+ 1=4, so the original formula = 4 (N2-2n+1) = 4&; Times; 4= 16.
Note: When merging similar terms in polynomials, we should be good at observing the overall characteristics of the problem and flexibly choose appropriate methods to answer it.
Example 4 is known: a-b=-3, b-c=2. Find the value of the algebraic expression (a-b)2+2(b-c)2-3(a-c)2.
Analysis: The value of algebraic expression (a-b)2+2(b-c)2-3(a-c)2 is required, and the values of a-b and b-c are given in the condition, not A, B and C, so the key to solve this problem is to know the value of A-c, and we can combine A-b and B-c.
Solution: Because a-b=-3 and b-c=2,
So (a-b)+(b-c)=- 1, that is, a-c=- 1.
When a-b=-3, b-c=2 and a-c=- 1,
(a-b)2+2(b-c)2-3(a-c)2 =(-3)2+2 & amp； Times; 22-3 & amp； Times; (- 1)2
=9+8-3。 Times; 1= 14.
Explanation: This question combines the similar items in two algebraic expressions by using the whole idea, so that the problem can be solved skillfully.
Example 5 shows that the algebraic expression 3a+4b has a value of 3. Find the value of algebraic expression 2(2a+b)+5(a+2b).
Solution: The original formula =4a+2b+5a+ 10b.
=9a+ 12b
=3(3a+4b)。
Therefore, when 3a+4b=3, the original formula =3(3a+4b)=9.