Chpt. 5, Polynomials and Polynomial Functions Flashcards
monomial
Either a real number, a variable, or a product of real numbers and variables with whole exponents.
degree of a monomial
The greatest degree out of all the monomials in the polynomial.
polynomial
A monomial or the sum of monomials.
degree of a polynomial
The greatest degree among its monomial terms.
polynomial function
A function that describes a polynomial.
standard form of a polynomial function
When the monomial terms are arranged by degree in descending order.
turning point
A point on the graph of a function where the graph changes direction, either from upwards to downwards, or from downwards to upwards.
end behavior
The end behavior of the graph of a function describes the behavior of the graph as the line moves away from the origin.
terms and polynomials can be named by their degree:
1st degree 2nd degree 3rd degree 4th degree 5th degree 6th degree
constant quadratic cubic quartic 5th degree 6th degree
factor theorem
The expression x-a is the factor of a polynomial function if and only if the value of “a” is the solution of the related polynomial function (since it’s the solution, it’s also the x-intercept).
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.
multiplicity
The number of times the related linear factor is repeated in the factored form of the polynomial.
relative maximum/minimum
The value of a function at a turning point on the graph.
sum of cubes
The sum of cubes is when you have a function of the form a^3 + b^3
It can be factored in the form:
(a +b) (a^2 -ab +b^2)
difference of cubes
The difference of cubes is when you have a function of the form a^3 -b^3
It can be factored in the form:
(a -b) (a^2 +ab +b^2)
synthetic division
A process for dividing a polynomial by a linear factor x -a. You list the standard form coefficients (including zeros) of the polynomial, omitting all variables and exponents. You use any factors of the greatest degree polynomial over any factors of the least degree polynomial. You add instead of subtract throughout the process.
remainder theorem
If a polynomial P(x) of degree n>1 is divided by x -a, the result is P(a).
conjugate root theorem
If P(x) is a polynomial with rational coefficients, then the irrational roots of P(x) = 0 occur in conjugate pairs. That is to say, if a +the root of b is a factor, then a -the root of b is by necessity also a factor. The same goes for imaginary numbers. If a - bi is a complex root with a and b real, then a +bi is also.
Descartes’ Rule of Signs
The number of positive real roots of P(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 P(-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 (count multiple roots according to their multiplicity).
fundamental theorem of algebra
If P(x) is a polynomial with a degree that is greater than or equal to 1 with complex coefficients, then P(x) = 0 has at least one complex root.
expand
To expand a binomial is to multiply it as needed until it is a polynomial in standard form.
Pascal’s Triangle
A triangular array of numbers in which the first and last number of each row are 1. Each of the numbers in the row is the sum of the two numbers above it.
binomial theorem
Pascal’s Triangle can be used to expand a binomial raised to a power. Whatever the power of the binomial is, go to that row of the triangle. Put each variable in front of Pascal’s constants (from the triangle). Finally, give one constant a degree starting with the degree of the binomial, and progressively becoming less until at the very end its zero. For the other variable, give it the opposite degree, in each term, as the first variable.
power function
A function of the form y = a * x^b, where a and b are nonzero real numbers.
constant of proportionality
If y = ax^b describes y as a power function of x, then y varies directly with (is proportional to) the bth power of x. The constant of a is the constant of proportionality.
