Flash cards
degree of a constant
The degree of a constant is 0. Ex: The degree of 5 is 0. The degree of -20 is 0.
difference of squares
a binomial that is the difference of two terms which are perfect squares Ex: (x + y) (x - y) = x^2 - y^2
factoring by grouping
a method of factoring that uses the distributive Property to remove a common binomial factor of two pairs of terms. Ex: x^3 + 3x^2 - 2x - 6 (x^3 + 3x^2) + (-2x - 6) x^2(x + 3) - 2(x + 3) (x + 3)(x^2 - 2)
constant
a monomial that does not contain a variable. Ex: The number -6 is the constant of the term 5x3 - 6
factors
a number or expression that is multiplied by another to yield a product. Ex: The factors of 2x2 - 5x - 3 are (2x + 1) and (x - 3) because when the binomials are multiplied, the result is 2x2 - 5x - 3. (2x + 1) (x - 3) 2x2 - 6x + x - 3 2x2 - 5x - 3
prime
a polynomial that cannot be factored any further; the only factors are 1 and itself Ex: 7x + 1, x2 + 5x + 1
binomial
a polynomial that contains 2 terms. Ex: 3x2 + 9
trinomial
a polynomial that has 3 terms Ex: 4x^2 + 7y - 15 1st 2nd 3rd
monomial
a polynomial that has only 1 term; an expression that is either a constant, a variable, or a product of numbers and variables Expressions like 2x + 3, 4 , or √m are NOT monomials. 2x + 3 is a sum, 4 is a quotient, and √m is a square root. Ex: 15, -7x2, x, and 3x3 y2
multiple(of a number)
a term that is the product of a given term and another term, or is a term that can be exactly divided by a given term
perfect square trinomial
a trinomial whose two binomial factors are identical a^2 + 2ab + b^2 x^2 + 2xy + y^2 (a + b)^2 (x + y)^2
polynomial
an expression whose terms are of the form axk, where k is a non-negative integer; a monomial or the sum or difference of two or more monomials Ex: 5x + 1, x2y + x, and 5 - 3c + 6c2
factored completely
decomposing a polynomial into factors which cannot be factored further Ex: The trinomial 2x2 - 5x - 3 yields factors (2x + 1) and (x - 3). The trinomial is factored completely because the factors 2x + 1 and x - 3 cannot be factored any further. When the trinomial 6x2 + 8x + 2 is factored into the binomials (3x + 1) and (2x + 2), the trinomial is not yet factored completely because the factor 2x + 2 can be rewritten as 2(x +1). The trinomial 6x2 + 8x + 2 completely factored: 2(x +1) (3x + 1)
term
each monomial in the polynomial Ex: _5xy^2_ - _3x_ + _5y^3_ -_ 3_
factoring a polynomial
expressing a polynomial as a list of factors, all of which, multiply to give the original polynomial Ex: The factors of 2x2 - 5x - 3 are (2x + 1) and (x - 3) because when the binomials are multiplied, the result is 2x2 - 5x - 3. (2x + 1) (x - 3) 2x2 - 6x + x - 3 2x2 - 5x - 3
standard form of a polynomial
terms are written in descending order, from the largest degree to the smallest degree Ex: y^2 + y^6 - 3y Find the degree of each term. Then arrange them in descending order. The standard form is y^6 + y^2 - 3y. The leading coefficient is 1.
leading coefficient
the coefficient of the 1st term of a polynomial written in standard form Ex: 5x^4 + 3x^3 - 7x^2 + 3x - 1 2b^2 + 5b^4 - 18b + 15b^7 + 6b^5 15b ^7 + 6b^5 + 5b^4 + 2b^2 - 18b
degree of polynomial
the largest degree of its terms. x^5 y^3z + 2xy^3 + 4x^2yz^2 5+3+1 1+3 2+1+2 =9 =4 =5
coefficient
the numerical factor when a term has a variable. The number a is the coefficient of the term axk. Ex: The number 5 is the coefficient of the term 5x3
degree of a monomial
the sum of the exponents of its variables. 2x^3 y^2 exponent of variable x is 3 exponent of variable y is 2 the degree of the monomial is 5 x^5 y^3z + 2xy^3 + 4x^2yz^2 5+3+1 1+3 2+1+2 =9 =4 =5
