Key Points

An algebraic identity is an equation that is true for all values of its variables, like $(x+y)^2 = x^2 + 2xy + y^2$. An equation is only true for specific values, for example $x+5=8$ is only true for $x=3$.

The identity for the square of a sum is $(a+b)^2 = a^2 + 2ab + b^2$. This is used for expanding expressions and simplifying calculations, such as finding $102^2 = (100+2)^2$.

The identity for the square of a difference is $(a-b)^2 = a^2 - 2ab + b^2$. It is useful for expanding expressions and finding squares of numbers like $99^2 = (100-1)^2$.

The identity for the difference of two squares is a^2 - b^2 = (a+b)(a-b). This is a key tool for factorization and for simplifying products like $103 \times 97 = (100+3)(100-3)$.

The product of two binomials with a common variable is given by the identity $(x+a)(x+b) = x^2 + (a+b)x + ab$. This is fundamental for factorizing quadratic trinomials.

To factorize a quadratic trinomial of the form $x^2 + px + q$, find two numbers $a$ and $b$ such that their sum is $a+b=p$ and their product is $ab=q$. The factors will then be $(x+a)(x+b)$.

The identity for the square of a trinomial is $(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$. Remember to include the three product terms.

The identity for the cube of a sum is $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. This can also be written in a factorized form as $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$.

The identity for the cube of a difference is $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$. This can also be written in a factorized form as $(a-b)^3 = a^3 - b^3 - 3ab(a-b)$.

The identity for the sum of two cubes is $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. This is used for factorizing expressions that are a sum of two perfect cubes.

The identity for the difference of two cubes is $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. This is another important factorization formula for the difference of two perfect cubes.

A key identity involving three variables is $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$.

A very useful result from the previous identity is that if $x+y+z=0$, then the identity simplifies to $x^3+y^3+z^3 = 3xyz$.

The identity for the product of two general binomials is $(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd$. This is useful for expanding and for factoring more complex trinomials.

To simplify a rational expression like \frac{P(x)}{Q(x)}, factorize both the numerator $P(x)$ and the denominator $Q(x)$ using suitable identities. Then cancel out any common factors, assuming the denominator is not zero.