VOCABULARY
Monomial –a real number, a variable, or a product of real numbers and variables with
whole number exponents
Degree of a monomial – the exponent of the variable if there is only one variable, or the
sum of the exponents if there is more than one variable
Polynomial – a monomial or the sum of monomials
Degree of a polynomial – the greatest degree among the monomial terms.
Polynomial function – a polynomial in the variable x
Standard form of a polynomial – arrange the terms by degree in descending numerical
order. A polynomial function P(x), where n is a nonnegative integer and 𝑎𝑎𝑛,… ,𝑎𝑎0 are real
numbers.
Turning point –a point where the graph of a function changes direction from upwards to
downwards or from downwards to upwards
End behavior – the direction of the graph of a function as you move to the left and to the
right, away from the origin.
Factor theorem – the expression (x – a) is a linear factor of a polynomial if and only if the
value of a is a root of the related polynomial function
Multiple zero –if a linear factor is repeated in the complete factored form of a polynomial,
the zero related to that factor is a multiple zero
EX. (𝑥)=2𝑥(𝑥−3)2(𝑥+1)
relative maximum – the y-value of the function at an up-to-down turning point.
relative minimum – the y-value of the function at an down-to-up turning point.
Sum of cubes – an expression in the form of 𝑎𝑎3+ 𝑏𝑏3. It can be factored as (𝑎+𝑏)(𝑎2−
𝑎+𝑏2)
EX. 𝑥3+8
Difference of cubes – an expression in the form 𝑎𝑎3− 𝑏𝑏3. It can be factored as
(𝑎−𝑏)(𝑎2+𝑎+𝑏2)
EX. 𝑥3−27
Division Algorithm for Polynomials – Divide polynomial (𝑥) by polynomial 𝐷(𝑥) to get
polynomial quotient 𝑄(𝑥) and polynomial remainder 𝑅𝑅(𝑥𝑥). The result is (𝑥)=𝐷(𝑥)𝑄(𝑥)+
𝑅(𝑥). If 𝑅(𝑥)=0, then 𝑃(𝑥)=𝐷(𝑥)𝑄(𝑥) and 𝐷(𝑥) and 𝑄(𝑥) are factors of 𝑃(𝑥).
Synthetic division – the process for dividing a polynomial by a linear expression x – a. A
polynomial must be in standard form. By omitting all variables and exponents
Remainder Theorem – If you divide a polynomial (𝑥) of degree 𝑛≥1 by x – a, then the
remainder is 𝑃(𝑎).
Rational Root Theorem – Any polynomial function with integer coefficients of the form
(𝑥)= 𝑎𝑛𝑥𝑛+ 𝑎𝑛−1𝑥𝑛−1+ …+ 𝑎1𝑥+ 𝑎0 has a limited number of possible roots of 𝑃(𝑥)=0.
Integer roots must be factors of 𝒂𝒂𝟎. Rational roots must have reduced form p/q
where p is an integer factor of 𝒂𝒂𝟎𝟎 and q is an integer factor of 𝒂𝒂𝒏𝒏.
Conjugate Root Theorem – If P(x) is a polynomial with rational coefficients, then the
irrational roots of (𝑥)=0 occur in conjugate pairs. That is, when (𝑎+ √𝑏) is a root, then (𝑎−
√𝑏) is also a root. Also, if 𝑎+𝑏 is a zero of a function, then 𝑎−𝑏 is also a zero.
Descartes’ Rule of Signs – Let P(x) be polynomial with real coefficients written in
standard form.
• The number of positive real roots of (𝒙)= 𝟎 is either equal to the number of sign
changes between consecutive coefficients of 𝑷(𝒙) or is less than that by an even
number.
• The number of negative real roots of (𝒙)= 𝟎 is either equal to the number of sign
changes between consecutive coefficients of 𝑷(−𝒙) or is less than that by an even
number.
Fundamental Theorem of Algebra – If (𝑥) is a polynomial of degree 𝑛≥1 with complex
coefficients, then 𝑃(𝑥)=0 has at least one complex root.
Example: (𝑥)=3𝑥3− 2𝑥+3 is a polynomial of degree 3, so 𝑃(𝑥)=0 has at least one complex
root.
Every quadratic polynomial equation has two roots, every cubic polynomial equation has
three roots, and so on.
expand – the operation of multiplying a binomial to a specific power, then writing the
polynomial in standard form.
Pascal’s Triangle – A rectangular array of numbers in which the first and last number is 1.
Each of the other numbers in the row is the sum of the two numbers above it.
Binomial Theorem – For every positive integer n, (𝑎+𝑏)= 𝑃0𝑎𝑛+ 𝑃1𝑎𝑛−1 𝑏+𝑃2𝑎𝑛−2 𝑏2+
…+ 𝑃𝑛−1𝑎𝑛−1+𝑃𝑛𝑏𝑛,
where 𝑃1,𝑃2, …,𝑃𝑛 are the numbers in the row of Pascal’s Triangle that has n as its second
number.