Key Points

A polynomial is an algebraic expression in which the exponents of the variables are only whole numbers. For example, $4x^2 - 3x + 7$ is a polynomial, but $3\sqrt{t} + t\sqrt{2}$ is not.

The degree of a polynomial is the highest power of the variable in the expression. For example, the degree of $x^5 - x^4 + 3$ is 5.

A polynomial with one term is a monomial (e.g., $5x^3$), two terms is a binomial (e.g., $x+1$), and three terms is a trinomial (e.g., $x^2 + x + \pi$).

A polynomial of degree one is linear ($ax+b$), degree two is quadratic ($ax^2+bx+c$), and degree three is cubic ($ax^3+bx^2+cx+d$).

A real number 'c' is a zero of a polynomial $p(x)$ if substituting 'c' for 'x' results in zero, that is, $p(c) = 0$. For $p(x) = x-2$, the zero is 2 because $p(2) = 0$.

A linear polynomial $ax+b$, where $a \neq 0$, has one and only one zero, given by the value $x = -\frac{b}{a}$.

If a polynomial $p(x)$ of degree greater than or equal to one is divided by the linear polynomial $x-a$, then the remainder is $p(a)$.

The expression $(x-a)$ is a factor of the polynomial $p(x)$ if and only if $p(a)=0$. This theorem is used to factorize polynomials of degree 2 or higher.

To factorize a quadratic polynomial $ax^2+bx+c$, we write $b$ as the sum of two numbers whose product is equal to $ac$.

Key identities are $(x+y)^2 = x^2+2xy+y^2$, $(x-y)^2 = x^2-2xy+y^2$, and $x^2-y^2 = (x+y)(x-y)$.

The square of a trinomial is given by the identity: $(x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx$.

The cube of a binomial sum is expanded as $(x+y)^3 = x^3+y^3+3xy(x+y)$ or $x^3+3x^2y+3xy^2+y^3$.

The cube of a binomial difference is expanded as $(x-y)^3 = x^3-y^3-3xy(x-y)$ or $x^3-3x^2y+3xy^2-y^3$.

Two important factorizations are $x^3+y^3 = (x+y)(x^2-xy+y^2)$ and $x^3-y^3 = (x-y)(x^2+xy+y^2)$.

A key identity for factorizing cubic expressions with three variables is $x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$.

A direct consequence of the previous identity is that if $x+y+z=0$, then the relationship simplifies to $x^3+y^3+z^3 = 3xyz$.