Algebra Part 3 Flashcards
What is a monomial?
An algebraic expression containing only one term.
Examples include expressions like 3x or 2y^2.
What does the Factor Theorem state?
If a polynomial is divided by (x - k) and the remainder is zero, then (x - k) is a factor of the polynomial.
This implies that k is a root of the polynomial.
What is a binomial?
An algebraic expression containing two terms.
Examples include expressions like x + y or 3a - 5b.
Define a polynomial.
An algebraic expression consisting of more than two terms.
Examples include expressions like x^3 + 2x^2 - x + 5.
What does Descartes’ Rule of Signs determine?
The maximum number of positive and negative real roots that a polynomial will have by counting the number of sign variations in the polynomial.
The polynomial must have real coefficients and be arranged in descending powers of x.
How can the number of positive real roots of a polynomial f(x) = 0 be determined?
It is either equal to the number of variations in sign of f(x) or less than that number by an even integer.
What is a sequence of numbers?
A set of numbers in a definite or specific order formed according to a definite rule. The numbers of the sequence are called terms.
What is the relationship between negative real roots and sign variations?
The number of negative real roots of a polynomial f(x) is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.
What is a progression?
The sequence of numbers called terms, each of which, after the first, is derived from the preceding one.
State the laws of exponents.
- a^0 = 1
- a^m * a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- a^(m/n) = √[n]{a^m}
What is an Arithmetic Progression (A.P.)?
A sequence of numbers where each term after the first is derived by adding a fixed number called the common difference.
What is the nth term of an A.P.?
a_n = a_1 + (n - 1)d
What is the formula for the sum of an A.P.?
S_n = n/2 * (a_1 + a_n)
What are the types of progression?
- Arithmetic Progression
- Geometric Progression
- Infinite Geometric Progression
- Harmonic Progression
What is the common difference in an A.P.?
The fixed number added to each term to derive the next term.
What is an algebraic equation?
A combination of real numbers and letters which represent numbers.
What is the formula for the Arithmetic Mean (A.M.)?
A_M = (a_1 + a_2 + … + a_n) / n
Fill in the blank: A binomial is an algebraic expression containing _______ terms.
two
True or False: A polynomial can only consist of one term.
False
What happens when a polynomial is divided by (x - r)?
The remainder is f(r).
