Chpt. 5, Polynomials and Polynomial Functions

Question 1

monomial

Answer 1

Either a real number, a variable, or a product of real numbers and variables with whole exponents.

Question 2

degree of a monomial

Answer 2

The greatest degree out of all the monomials in the polynomial.

Question 3

polynomial

Answer 3

A monomial or the sum of monomials.

Question 4

degree of a polynomial

Answer 4

The greatest degree among its monomial terms.

Question 5

polynomial function

Answer 5

A function that describes a polynomial.

Question 6

standard form of a polynomial function

Answer 6

When the monomial terms are arranged by degree in descending order.

Question 7

turning point

Answer 7

A point on the graph of a function where the graph changes direction, either from upwards to downwards, or from downwards to upwards.

Question 8

end behavior

Answer 8

The end behavior of the graph of a function describes the behavior of the graph as the line moves away from the origin.

Question 9

terms and polynomials can be named by their degree:

1st degree
2nd degree
3rd degree
4th degree
5th degree
6th degree

Answer 9

constant
quadratic
cubic
quartic
5th degree
6th degree

Question 10

factor theorem

Answer 10

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).

Question 11

multiple zero

Answer 11

If a linear factor is repeated in the complete factored form of a polynomial, the zero related to that factor is a multiple zero.

Question 12

multiplicity

Answer 12

The number of times the related linear factor is repeated in the factored form of the polynomial.

Question 13

relative maximum/minimum

Answer 13

The value of a function at a turning point on the graph.

Question 14

sum of cubes

Answer 14

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)

Question 15

difference of cubes

Answer 15

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)

Question 16

synthetic division

Answer 16

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.

Question 17

remainder theorem

Answer 17

If a polynomial P(x) of degree n>1 is divided by x -a, the result is P(a).

Question 18

conjugate root theorem

Answer 18

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.

Question 19

Descartes’ Rule of Signs

Answer 19

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).

Question 20

fundamental theorem of algebra

Answer 20

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.

Question 21

expand

Answer 21

To expand a binomial is to multiply it as needed until it is a polynomial in standard form.

Question 22

Pascal’s Triangle

Answer 22

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.

Question 23

binomial theorem

Answer 23

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.

Question 24

power function

Answer 24

A function of the form y = a * x^b, where a and b are nonzero real numbers.

Question 25

constant of proportionality

Answer 25

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.