Polynomials

Polynomials enable us to express relationships between constants and indeterminates. These expressions can then be manipulated through various algebraic methods.

Polynomial Terms

Some important terms include...

1. A polynomial is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. This means it is an expression that can be writen in the form $a_nx^n + ... + a_2x^2 + a_1x + a_0$.
2. A coefficient is the real number that is multiplied by variable raised to an exponent.
3. A constant is a number that is not multiplied by a variable. This technically is the coefficient to $x^0$.
4. A term of a polynomial is any product term in the polynomial.
5. The leading term is the term with highest power.
6. The leading coefficient is the coefficient of the leading term.
7. The degree of a polynomial is the power of the leading term.

Types of Polynomials

The types of polynomials include...

1. A monomial is a polynomial containing only one term ($ex.\: 7x^3$).
2. A binomial is a polynomial containing two terms ($ex.\: 5x + 14$).
3. A trinomial is a polynomial containing three terms ($ex.\: -9x^2 + 4x - 5$).

Manipulating Polynomials

We add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. ($ex.\: 3x^3$ and $-9x^3$ are like terms but $3x^2$ and $3x^9$ are not like terms).

To multiply polynomials, we use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. An example is...

$\underline{Example}$

$(4x - 1)(2x^2 + 2x + 1)$

$4x (2x^2 + 2x + 1) - 1 (2x^2 + 2x + 1)$ Use the distribute property

$(8x^3 + 8x^2 + 4x)$ + $(-2x^2 - 2x - 1)$ Multiply

$8x^3 + (8x^2 - 2x^2) + (4x - 2x) - 1$ Combine like terms

$8x^3 + 6x^2 + 2x - 1$ Simplify

Binomials

The FOIL method is a method we can use to find the product of two binomials. It involves us multiplying the first terms, the outer terms, the inner terms, and then the last terms of each binomial. This means $(ax + b)(cx + d) = acx^2 + adx + bcx + bd$.

Common binomial products include...

Name	Definition
Perfect Square Trinomials	$(x + a)^2 = (x + a)(x + a) = x^2 + 2ax + a^2$
Difference of Squares	$(a + b)(a - b) = a^2 - b^2$

note

A polynomial can contain several variables. An expression like $3x - 2y + 5$ is also a polynomial.