The polynomial is an expression consisting of the amount of single-wing. The latter are the product of the constant (numbers) and the root (or roots) of the expression to the degree k. In this case, they speak of a polynomial degree K. The decomposition of the polynomial implies the transformation of the expression in which multipliers come to the change of terms. Consider the main ways to carry out this kind of transformation.
Method of decomposition of polynomial by highlighting a common factor
This method is based on the laws of the distribution law. So, Mn + Mk \u003d M * (n + k).
- Example:spread 7Y 2.+ 2UY and 2M 3- 12m 2 + 4lm.
7Y. 2.+ 2UY \u003d Y * (7Y + 2U),
2m. 3- 12m 2 + 4lm \u003d 2m (m 2- 6m + 2L).
However, the multiplier present in each polynomial may not always be found, therefore this method is not universal.
Method of decomposition of polynomial based on formulas of abbreviated multiplication
The formulas of the abbreviated multiplication are valid for a polynomial of either degree. In general, the transformation expression is as follows:
u. k.- L. k.\u003d (U - L) (U k-1 + U. k-2* L + U k-3.* L. 2+ ... u * L k-2+ L. k-1), where K is a representative of natural numbers.
Most often in practice, formulas for polynomials of the second and third orders are used:
u. 2- L. 2.\u003d (U - L) (U + L),
u. 3- L. 3.\u003d (U - L) (U 2.+ UL + L 2.),
u. 3+ L. 3\u003d (U + L) (U 2 - Ul + L 2.).
- Example:spread 25p 2- 144b. 2.and 64m. 3- 8L 3.
25p 2- 144b. 2\u003d (5p - 12b) (5p + 12b),
64m 3- 8L 3\u003d (4m) 3- (2L) 3\u003d (4m - 2L) ((4m) 2+ 4m * 2L + (2L) 2) \u003d (4m - 2L) (16m 2 + 8ml + 4L 2).
Method of decomposition of polynomial - grouping of terms of expressions
This method will in some way echoes with the technique of removing a common factor, but has some differences. In particular, before selecting a common factor, a grouping of universes should be made. The basis of the grouping is the rules of combinating and moving laws.
All are unarranged, presented in the terms are divided into groups, in each of which the general value is made such that the second factor will be the same in all groups. In general, a similar method of decomposition can be represented as an expression:
pL + KS + KL + PS \u003d (PL + PS) + (KS + KL) ⇒ PL + KS + KL + PS \u003d P (L + S) + K (L + S),
pL + KS + KL + PS \u003d (P + K) (L + S).
- Example:spread 14mn + 16LN - 49M - 56L.
14mn + 16LN - 49M - 56L \u003d (14mn - 49m) + (16LN - 56L) \u003d 7M * (2N - 7) + 8L * (2N - 7) \u003d (7m + 8L) (2N - 7).
Method of decomposition of polynomial - Forming a full square
This method is one of the most effective during the decomposition of the polynomial. At the initial stage, it is necessary to determine the names that can be "collapse" into the square of the difference or amount. To do this, uses one of the relations:
(P - B) 2.\u003d P. 2.- 2PB + B 2,
(P + B) 2.\u003d P. 2.+ 2PB + B 2.
And then convert a polynomial based on the formulas of abbreviated multiplication.
- Example: Decide the expression U. 4+ 4U. 2 - 1.
We highlight among its homoral terms, which form a full square: u 4+ 4U. 2 - 1 \u003d u 4+ 2 * 2U 2 + 4 - 4 - 1 \u003d
\u003d (U. 4+ 2 * 2U 2 + 4) - 4 - 1 \u003d (U 4+ 2 * 2U 2 + 4) - 5.
Next, turn the expression in brackets according to the full square formula: (U 4+ 2 * 2U 2 + 4) - 5 \u003d (U 2+ 2)2– 5.
Complete the transformation using abbreviated multiplication rules: (U 2+ 2)2- 5 \u003d (U 2+ 2 - √5) (U 2+ 2 + √5).
That. U. 4+ 4U. 2 - 1 \u003d (U 2+ 2 - √5) (U 2+ 2 + √5).
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