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polynomial

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An expression that can be written in the form shown in the image where all the coefficients are real numbers and all the exponents are positive integers.

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degree of a Polynomial

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greatest degree (exponent) of all the monomials

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constant

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degree of 0 (example: 7)

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linear

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degree of 1 (example: 2x)

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quadratic

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degree of 2 (example: -3x²+2)

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cubic

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degree of 3 (example: -4x³-2x)

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quartic

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degree of 4 (example: x⁴+2x)

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monomial

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polynomial with one term

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Trinomial

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polynomial with three terms

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standard Form

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a polynomial whose terms are ordered by degree from greatest to least

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factored form

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a polynomial expression written as a product of factors. Example: 2x(2x-3)(x+5)

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binomial

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polynomial with two terms

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A Closed Set Under and Operation

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Performing the operation on two members of the set produces another member of the set.

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Imaginary number

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A number that includes a i, which is the square root of -1

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greatest common factor

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The expression with the largest number and highest degree that can be divided evenly into each term of an expression

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Factoring by grouping

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Factoring by pulling out the GCF of groups of terms

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factoring by substitution

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Factoring by substituting an expression with a "u" so that the expression becomes quadratic

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turning point

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A point at which the polynomial function values change from increasing to decreasing or decreasing to increasing.

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relative minimum

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A y-value that is smaller than the values for all other y-values in some neighborhood in the domain of a function.

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relative maximum

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A y-value that is larger than the values for all other y-values in some neighborhood in the domain of a function.

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difference of squares

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x²-y² = (x+y)(x-y)

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square of a binomial

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x²±y² = x²+2xy+y²

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difference of cubes

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a³-b³=(a-b)(a²+ab+b²)

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sum of cubes

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a³ + b³ = (a + b)(a² - ab + b²)

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binomial expansion

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A method of expanding the expression (x+y)ⁿ for any whole number, n.

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multiplicity

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the number of times that a factor is repeated in the factorization of a polynomial expression

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Fundamental Theorem of Algebra

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A polynomial, P(x), of degree n≥1 has exactly n total complex roots (repeated roots counted separately)

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Rational Roots Theorem

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States a constraint on rational solutions of a polynomial equation.
The theorem states that each rational solution x = p/q, written in lowest terms so that p and q are relatively prime, satisfies:
p is an integer factor of the constant term a0, and
q is an integer factor of the leading coefficient an.

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Remainder Theorem

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if the polynomial expression that defines the function P is divided by x-a then the remainder is the number P(a)

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Factor Theorem

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c is a zero of polynomial P(x) if and only if x-c is a factor of P(x).

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end behavior

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describes the values of a function at the positive and negative extremes in its domain

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Turning Point Theorem

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If f is a polynomial of degree n, then f has at most n-1 turning points. Conversely, If the graph of a polynomial function has n-1 turning points, then the function has a degree of n or more.

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power function

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A function of the form f(x) = axⁿ , where n is a real number.

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Intermediate Value Theorem (IVT)

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If f is continuous on the closed interval [a,b]
and k is between f(a) and f(b), then there exists a number c between a and b for which f(c) = k. (Note that this is an existence theorem.)