Trimester II Vocabulary (Chapters 5, 6, and 8)

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Factors

Numbers that multiply together to get a product.

Greatest Common Factor (GCF)

The largest integer that is a factor of all the integers in a list.

Factoring

The process of writing a polynomial as a product.

GCF of a List of Common Variables
Raised to Powers

The variable raised to the smallest exponent in the list.

GCF of a List of Terms

The product of all comon factors.

Prime Factorization

The process of writing a number as a product of its prime factors.

Factoring by Grouping

A technique for factoring a four-term polynomial where you group terms in two groups of two terms so that each group has a common factor, factor out the GCF of each group, and then factor out a common binomial.

Factoring a Trinomial in the Form
x² + bx + c

The factored form is (x + #1)(x + #2),
where #1 and #2 have a product of c and a sum of b.

Prime Polynomial

A polynomial that is not factorable with intergers.

Factoring a Perfect Square Trinomial

A trinomial that is the square of some binomial can be factored as follows:
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²

Factoring a Trinomial in the Form
ax² + bx + c

The process of trying various combinations of factors of 'ax²' and 'c' until a middle term of 'bx' is obtained when checking.

Factoring a Trinomial in the Form
ax² + bx + c by Grouping

The process of finding two numbers whose product is 'a' times 'c' and whose sum is 'b', rewriting 'bx' using the factors found and then factoring by grouping to finish.

Factoring a Difference of Squares

A binomial that is a difference of squares can be factored as follows:
a² - b² = (a + b)(a - b)

Factoring the Sum of Two Cubes

A binomial that is a sum of cubes can be factored as follows:
a³ + b³ = (a + b)(a² - ab + b²)

Factoring the Difference of Two Cubes

A binomial that is a difference of cubes can be factored as follows:
a³ - b³ = (a - b)(a² + ab + b²)

Quadratic Equation in Standard Form

An equation that can be written in the form:
ax² + bx + c = 0
where a, b, and c are real numbers and a ≠ 0.

Zero Factor Theorem

If a and b are real numbers and if ab = 0, then
a = 0 or b = 0.

Factoring a Quadratic Equation

The process of writing an equation in standard form so that one side of the equation is 0, factoring the quadratic, setting each factor containing a variable equal to 0 and solving.

Index

The 4 in ∜16.

Radicand

The 16 in ∜16.

Radical

The symbol √.

Square Root

If 'a' is a positive number, then
√a = b if b² = a

Cube Root

Written as ∛a; if 'a' is a real number,
then ∛a = b only if b³ = a

Product Rule for Radicals

If √a and √b are real numbers,
then √a ∙ √b = √(a ∙ b), for any index

Quotient Rule for Radicals

If √a and √b are real numbers and b ≠ 0,
then √(a / b) = √a / √b, for any index

Like Radicals

Two or more radical expressions that have the same index and the same radicand.

Rationalizing the Denominator

The process of eliminating the radical in the denominator of a radical expression

Conjugate

Used to rationalize a denominator for a sum or difference of radicals. The sign in between becomes the opposite (addition to subtraction, or subtraction to addition). For example, a + b becomes a - b.

Squaring Property of Equality

If a = b, then a² = b².

Pythagorean Theorem

If a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse, then a² + b² = c².

Distance Formula

The distance d between two points with coordinates
( x₁, y₁) and ( x₂, y₂) is given by
d = √( x₂ - x₁)² + ( y₂ - y₁)².

Extraneous Solution

When a proposed solution is not an actual solution.

Complex Number

A number written in the standard form a + bi.

Pure Imaginary Number

A complex number written as 0 + bi.

Pure Real Number

A complex number written as a + 0i.

Imaginary Number

Written as i, is the number whose square is -1.

Rational Exponent

An exponent that can be written as a fraction.

a^(1/n)

The nth root of a.

Term

A number, variable, or product of both.

Zero Exponent

Any quantity raised to this exponent equals one.

Polynomial

A finite sum of terms of the form ax^n, where a is a real number and n is a whole number.

Monomial

A polynomial with exactly one term.

Binomial

A polynomial with exactly two terms.

Trinomial

A polynomial with exactly three terms.

Degree of a Term

The sum of the exponents on the variables in the term. Ex: -5x²y³ -> degree is 5.

Degree of a Polynomial

The greatest exponent of any term of the polynomial. Determines how many times the polynomial could cross the x-axis.

Coefficient

The numerical factor of a term.

Constant

The number term of a polynomial.

Double Distribute Method

Method used to multiply two binomials.

Dividend

The number that gets divided.

Divisor

The number that is used to go into the dividend.

Quotient

The result of a division problem.

Leading Coefficient

The coefficient of the term with the highest degree.

Constant

Degree of 0; y=3

Linear

Degree of 1; y=2x + 3

Quadratic

Degree of 2; y = 2x² + 3

Cubic

Degree of 3; y=2x³ +3x + 3

Conjugates

Binomials with the same terms, one with a plus, the other with a minus; (x+y)(x-y).

Odd Degree Polynomials

Polynomials that MUST cross the x-axis at least once.

Even Degree Polynomials

Polynomials that do not have to cross the x-axis, but could.

Multiplying the Sum and Difference of Two Terms

(a + b)(a - b) = a² - b²

Dividing by a Monomial

Divide each term of the polynomial in the numerator by the monomial: (a + b + c)/d = a/d + b/d + c/d.

Long Division

Necessary in division polynomial problems when the divisor has two or more terms.

Synthetic Division

Able to use in division polynomial problems when the divisior is a linear binomial.

Scientific Notation

A positive number is written as a product of a number 'a', where 1≤ a <10, and an integer of power 'r' of 10:
a x 10^r.

Simplify a Polynomial

Combine like terms, reduce coefficients, and make sure there are no negative exponents.

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