10.2 using combinations and the binomial theorem

This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.

The document provides an overview of a lecture covering matrices, matrix algebra, vectors, homogeneous coordinates, and transformations in homogeneous coordinates. Key points include: matrices are arrays of numbers; operations on matrices include addition, multiplication by a scalar, and multiplication; vectors can represent points in space as column or row matrices; homogeneous coordinates allow points to be represented by 4D vectors, enabling translations and other transformations to be described by 4x4 matrices. This provides a unified approach for combining multiple transformations.

The document discusses inverse functions, including: - An inverse function undoes the output of the original function by relating the input and output variables. - For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input. - To find the inverse of a function, swap the input and output variables and isolate the new output variable.

This document provides instructions for using synthetic division to divide polynomials. It contains the following key points: 1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator. 2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder. 3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15

1) This document discusses polynomial operations and rules including combining like terms, adding, subtracting, and multiplying polynomials. Key terms defined include degree of a polynomial, standard form, leading coefficient, monomial, binomial, trinomial, and polynomial. 2) Examples are provided for combining like terms, adding, subtracting, and multiplying polynomials using the distributive property. Special cases like FOIL (First, Outer, Inner, Last) are explained for multiplying binomials. 3) Practice problems with answers are given for multiplying polynomials.

This document discusses permutation and combination concepts in counting. It provides examples of using the multiplication principle when order matters and addition principle when order does not matter to count outcomes of events. It also defines permutation as arrangements of objects where order matters and combination as selections of objects where order does not matter. Formulas are given for counting permutations and combinations of distinct objects. Examples demonstrate calculating the number of ways to arrange or select objects in different scenarios.

The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.

Long division can be used to divide polynomials in a similar way to dividing numbers. The key steps are to set up the division problem, divide the term of the dividend by the term of the divisor, multiply the divisor by the quotient term and subtract, then bring down the next term of the dividend and repeat. This polynomial long division allows polynomials to be factored by finding all divisor polynomials that give a remainder of zero. The factor theorem can also be used to check if a linear polynomial is a factor by setting it equal to zero and checking if it makes the other polynomial equal to zero.

The document contains solutions to 3 probability problems: 1) Finding the probability of rolling a die with a face that is a multiple of 3 or 5. The probability is 5/6. 2) Finding the probability that a randomly selected digit between 1-50 is a multiple of 10 or 11. The probability is 18/50. 3) Finding the probability that a randomly drawn card is a queen or king. The probability is 8/52.