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How to factor

how to factor

Spread integer and polynomial factorization.

Remembering school technique of long division.



Any integer can be decomposed into prime factors.
To do this, you must share it consistentlyby numbers, starting with 2. And it may happen that some numbers will be included in the expansion of more than once. That is, by dividing the number by 2, do not rush to move on to the top three, again, try to divide it into two.

And here we will help signs of divisibility: 2 even numbers divided by 3 the number of shares, if the sum of digits contained in it is divided into three, divided by 5 numbers ending in 0 and 5.

Share best in a column. Starting with the left digit number (or two left digits) sequentially divide the number by the corresponding factor, the result is recorded in the particular. Next, multiply the intermediate quotient by the divisor and subtract from a selection of the dividend. If the number is divided by its putative prime factor, the balance is supposed to be zero.

Example of a polynomial expansion into factors


A polynomial can also be factored.

There are various approaches: You can also try to group the terms, you can use the well-known formulas of abridged multiplication (the difference of the squares, the square sum / difference, cubic sum / difference of cubes difference).

You can also use the method of selection: if the number has come as a solution, it is possible to divide the original polynomial expression in the (x- (number is found)) you have chosen. For example, a column. Polynomials divided evenly, and it decreases by one degree. It must be remembered that the degree of the polynomial P has no more than P of different roots, but the roots may be the same, so try to substitute the numbers found above in the simplified polynomial - it is possible that long division can be repeated again.

It writes the result obtained as a product of expressions of the form (x (1 root)) * (x (root 2)) ... etc.

Formula skoraschёnnogo multiplication

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