Polynomials-Types, Graphs, factor and Remainder theorem
Some Algebraic Identities
1.\(
(a + b)^2 = a^2 + 2ab + b^2
\)
2.\(
(a - b)^2 = a^2 - 2ab + b^2
\)
3.\(
(a + b)^2 + (a - b)^2 = 2(a^2 + b^2 )
\)
4.\(
(a + b)^2 - (a - b)^2 = 4ab
\)
5.\(
(a + b)(a - b) = a^2 - b^2
\)
6.\(
(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b)
\)
7.\(
(a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a - b)
\)
8.\(
a^3 + b^3 = (a + b)(a^2 - ab + b^2 )
\)
9.\(
a^3 - b^3 = (a - b)(a^2 + ab + b^2 )
\)
10.\(
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
\)
11.\(
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
\)
12.If \(
a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc
\)
13.\(
a^k + b^k = (a^{k - 1} + b^{k - 1} )(a + b) - (a^{k - 2} + b^{k - 2} )ab
\)
Special Products
1. \(
(x + a)(x + b) = x^2 + (a + b)x + ab
\)
2. \(
(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd
\)
3. \(
(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc
\)