Polynomials
Expressions built from a variable by adding and multiplying behave a lot like whole numbers. How do they add, multiply, factor and divide?
Where you are 10 inside
What this area is about
A polynomial is a sum of terms, each a number times a whole-number power of the variable, such as 2x^3 + 3x^2 + x. Polynomials add, multiply and divide much as integers do, and factoring one into simpler polynomials is like factoring a number into primes. This area does that arithmetic and ends with what one evaluation, p(c), says about factors and roots.
How these fit together
The object. Degree, terms and standard form names the parts; the degree is the highest power.
Operations. Adding and subtracting polynomials combines like terms. Multiplying polynomials multiplies every term by every term. Special products are the squared binomial and sum times difference, recognized on sight.
Factoring. Factoring trinomials reverses the expansion of two binomials. Factoring special patterns reads the special products backward, as a^2 - b^2 = (a + b)(a - b), and adds the sum and difference of two cubes.
Division. Polynomial long division works one power at a time; synthetic division is its shorthand when the divisor is x - c.
Results. The remainder and factor theorems link p(c), the remainder and the roots. Proving polynomial identities shows that two expressions agree for every value.
A path through it
Degree first; then adding and multiplying. After multiplying: factoring trinomials and long division. Special products come before both special patterns and identities. Trinomials and special patterns refer to each other, so take them together. Long division, then synthetic division, then the theorems.
After this: quadratics, rational expressions and polynomial functions.