Polynomials
Polynomials: Sums and Differences
A term is an indicated product and may contain any number of factors. The variables in a term are called literal factors, and the numerical factor is called the numerical coefficient.
The degree of a monomial is the sum of the exponents of the literal factors.
Definition
A polynomial is a monomial or a finite sum (or difference) of monomials.
The degree of a polynomial is the degree of the term with the highest degree
Products and Quotients of Monomials
Property 3.1 Product of the Same Base with Integer Exponents
If \(b\) is any real number, and \(n\) and \(m\) are positive integers, then
$$ b^n \cdot b^m = b^{m + n} $$Property 3.3 Power Raised to a Power
If \(b\) is any real number, and \(n\) and \(m\) are positive integers, then
$$ (b^n)^m = b^{mn} $$Property 3.3 Power of a Product
If \(b\) is any real number, and \(n\) is a positive integer, then
$$ (ab)^n = a^n b^n $$Dividing Monomials
Property 3.4 Quotient of Same Base with Integer Exponents
If \(b\) is any nonzero real number, and \(n\) and \(m\) are positive integers, then
$$ \frac{b^n}{b^m} = b^{n - m}, \text{ when n > m } $$$$ \frac{b^n}{b^m} = 1, \text{ when n = m } $$Factoring: Greatest Common Factor and Common Binomial Factor
Definition
If a positive integer greater than \(1\) has no factors that are positive integers other than itself and \(1\), then it is called a prime number.
Definition
A positive integer greater than \(1\) that is not a prime number is called a composite number.
The indicated product form that contains only prime factors is called the prime factorization form of a number.
A polynomial with integral coefficients is in completely factored form if:
- It is expressed as a product of polynomials with integral coefficients
- No polynomial, other than a monomial, within the factored form can be further factored into polynomials with integral coefficients.
Property 3.5
Let \(a\) and \(b\) be real numbers. Then:
\(ab = 0\) is and only if \(a = 0\) or \(b = 0\)
Factoring: Difference of Two Squares and Sum or Difference of Two Cubes
Identity Difference of Two Squares
We say that a polynomial is a prime polynomial if it is not factorable using integers.
Factoring the Sum and Difference of Two Cubes
Identity Sum and Difference of Two Cubes