Decomposition of factoring problems in the second year of junior high school

a^2×(b+c)+b^2×(a+c)+c^2×(a+b)+2abc

Idea analysis: Observe the structural characteristics of the above formula.

The positions of b,c are the same, i.e., the positions of a,b,c are interchangeable everywhere for the value identity of the above formula. This is mathematically called symmetry.

2.Each item is triplicated.

The number of factors should be less than three, and the items that meet feature 1 are.

a^2+b^2+c^2

a+b+ca+b)(b+c)(c+a)

a-b)(b-c)(c-a)

This category is the most basic symmetric rotational polynomial.

Combine the above two features and the original formula to make a bold guess only.

a^2×(b+c)+b^2×(a+c)+c^2×(a+b)+2abc

k(a^2+b^2+c^2)(a-b)(b-c)(c-a)

where k=1a2 (b+c)+b2(a+c)+c2(a+b)+2abc

a^2+b^2+c^2)(a-b)(b-c)(c-a)

Symmetry is very universal in mathematics and is one of the five principles of mathematical aesthetics. Here it is represented as a symmetrical rotation polynomial, which is also very common in this polynomial, so I won't say much about it.

To be continued:**Wait for 5 minutes, I have to continue to revise and improve my reply, if the final reply time is more than 5 minutes, or the number of revisions reaches the upper limit of 10 times and cannot continue to modify, it automatically means that I have finished answering.

If you want to be the problem finisher, you have to do no or less useless work, which can only be carefully served for you, don't be annoyed by it.

You didn't write it right, you can't see what you're talking about! I'm sorry!

For example: 2a(a-b)+5b(a-b).

Solution: 2a(a-b)+5b(a-b)=(a-b)*(2a+5b) (extract common factor).

Such as: a 2-4b 2

Solution: a 2-4b 2 = (a + 2b ) (a - 2b ) (square difference formula).

Because. a4+b2c2=b4+a2c2

So. a4+b2c2-b4-a2c2=0

a4-b4+b2c2-a2c2=0

a2+b2)(a2-b2)+c2(b2-a2)=0(a2+b2)(a2-b2)-c2(a2-b2)=0(a2+b2-c2)(a+b)(a-b)=0 When a2+b2-c2=0, abc is a right triangle.

When a-b=0, abc is an isosceles triangle.

When a2+b2-c2=0 a-b=0, abc is an isosceles right triangle, but to be honest, this question is a bit weird

Mention the common factor 4x 2: (8x to the power of 12x) (x-3) = 4x 2 (2x-3) (x-3).

Think of 2a+b as a number, using the perfect square formula, the square of (2a+b+1) = 0

2a+b=minus 1, then the answer is 1

x²+y²-10x+8y+45

x²-10x+25+y²+8y+16+4=(x-5)²+y+4)²+4

Because (x-5) >=0 ,(y+4) >=0, (x-5) +y+4) +4>0

That is, no matter what real number x and y take, the value of the polynomial x + y -10x + 8y + 45 is always positive.

x²+y²-10x+8y+45=（x-5)²+y+4)²+4≥4>0

Therefore, the original formula is greater than or equal to 4, that is, greater than 0, so no matter what real number x and y are taken, the value of the polynomial x + y -10x + 8y + 45 is always positive.

x +y -10x+8y+45=(x -10x+25)+(y +8y+16)+4=(x-5) +y+4) +4 Evergrande is 0

Regardless of the real number taken by x and y, the value of the polynomial x + y -10x + 8y + 45 is always positive.