MENUPolynomial Identities
When we have a sum(difference) of two or three numbers to power of 2 or 3 and we need to remove the brackets we usepolynomial identities(short multiplication formulas):
(x - y)2 = x2 - 2xy + y2
Example 1: If x = 10, y = 5a
(10 + 5a)2 = 102 + 2·10·5a + (5a)2 = 100 + 100a + 25a2
Example 2: if x = 10 and y is 4
(10 - 4)2 = 102 - 2·10·4 + 42 = 100 - 80 + 16 = 36
The opposite is also true:
25 + 20a + 4a2 = 52 + 2·2·5 + (2a)2 = (5 + 2a)2
Consequences of the above formulas:
(-x - y)2 = (-(x + y))2 = (x + y)2 = x2 + 2xy + y2
Formulas for 3rd degree:
(x - y)3 = x3 - 3x2y + 3xy2 - y3
Example: (1 + a2)3 = 13 + 3.12.a2 + 3.1.(a2)2 + (a2)3 = 1 + 3a2 + 3a4 + a6
(x - y - z)2 = x2 + y2 + z2 - 2xy - 2xz + 2yz
Factor Rules
x2 + y2 = (x + y)2 - 2xy
or
x2 + y2 = (x - y)2 + 2xy
Example: 9a2 - 25b2 = (3a)2 - (5b)2 = (3a - 5b)(3a + 5b)
x3 + y3 = (x + y)(x2 - xy + y2)
If n is a natural number
If n is even (n = 2k)
If n is odd (n = 2k + 1)
More Algebraic Formulas
(a + b)2 - (a - b)2 = 4ab
(a - b)2 = (a + b)2 - 4ab
a4 - b4 = (a + b)(a - b)[(a + b)2 - 2ab]
Problems Involving Polynomial Identities
1) Simplify the expression
$(a - b)^2 - 2(a - b)(a + b) + (a + b)^2 =$
Solution:
$a^2 - 2ab + b^2 - 2(a^2 - b^2) + a^2 + 2ab + b^2 = 2a^2 + 2b^2 - 2a^2 + 2b^2 = 4b^2$
2) Simplify the expression
$(x^2 + 2)^2 - (x - 2)(x + 2)(x^2 + 4)=$
Solution:
$x^4 + 4x^2 + 4 - (x^2 - 4)(x^2 + 4)=x^4 + 4x^2 + 4 - (x^4 - 16) = x^4 + 4x^2 + 4 - x^4 + 16 = 4x^2 + 20$
3) Solve the equation: x2 - 25 = 0
Solution: x2 - 25 = (x - 5)(x + 5)
=> we have to solve the following 2 equations:
x - 5 = 0 or x + 5 = 0
so the equation have two decisions: x = 5 and x = -5
Related Resources:
Simplifying polynomial expressions - problems with solutions
