Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler components, typically by finding its factors. This process is important in algebra and calculus, as it helps solve equations, find roots, and simplify expressions. Factoring can be done using various methods, such as the distributive property, grouping, difference of squares, and more.
3 Key excerpts on "Factoring Polynomials"
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...For example, because –5 divides evenly into each term in the numerator. If the coefficient of the x 3 term had been a 7, then we wouldn’t have been able to do this division so smoothly. By the way, the reason we say most (and not all) bets are off is that it can be done, but we’ll leave that to the mathematicians. Just as factoring was important when dividing any two quantities, whether they are numbers, fractions, or anything else, factoring is an important part of dividing two polynomials. For example, by recognizing that there is a common factor (x) in the numerator and denominator of we get In fact, in this example, the numerator can be factored further, and the whole expression equals (x + 5). But we are getting ahead of ourselves a little here. First we must talk about quadratic equations and how to factor them, the topic of the next section. Quadratic Equations A quadratic equation is an equation in which the unknown is squared and there is no higher power of the unknown. It is okay if there are no lower powers of the unknown (in other words, no “x” term or no “pure number” term). An example of a quadratic equation is x 2 + x –6 =0, and so are x 2 – 9 = 0 and x 2 +3 x = 0. Quadratic equations always have two answers for the value of the unknown (even though at times they are the same number twice). The general form of a quadratic equation is ax 2 + bx + c = 0, where b and c can be any numbers, even 0, as we saw with the examples of x 2 –9 = 0 and x 2 +3 x = 0. If a = 0, though, we no longer have a quadratic–according to the definition, there has to be a squared term. Solving a Quadratic Equation The solutions to a quadratic equation are based on a simple fact: if two factors are multiplied together and the product is 0, then either one or both of the factors must equal 0. There just are no two nonzero numbers whose product is 0. Period...
eBook - ePub- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Unless noted otherwise, when factoring a polynomial, we want factors with integer coefficients. If a polynomial cannot be factored, it is considered to be prime. To factor a trinomial in the form ax 2 + bx + c, we use the following pattern, which is a reverse of the FOIL method: The goal is to find a combination of factors of a and c such that the outer and inner products add up to the middle term bx. According to the particular polynomial, determine which method below to take to make the process somewhat easier. Method 1: Use this one when a = 1 and the sign of c is positive. This is the easiest case. The factors of the polynomial both have the same sign as the coefficient of b, their sum equals the coefficient of b, and their product equals c. For example, to find the factors of x 2 + 7 x + 10, look at the factors of 10 (the value of c) to see which have a sum of 7 (the value of b). These are 5 and 2, so the factors are (x + 2)(x + 5). Method 2: Use this when a = 1 and the sign of c is negative. The factors of the polynomial have different signs. Again, their sum equals the coefficient of b, and their product equals c. For example, to find the factors of x 2 − 3 x − 4, look at the factors of −4 (the value of c) to see which have a sum of −3 (the value of b). These are −4 and 1, so the factors are (x − 4)(x + 1). Method 3: Use this when a ≠ 1. Write the whole number factors of a in one row and the integer number factors of c in another row, and check which combination adds to b. This can be very tedious because there can be many combinations to check. However, once we have found one that works, we don’t have to continue checking the others...
eBook - ePubFactor Analysis
Classic Edition
- Richard L. Gorsuch(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...3 MATRIX ALGEBRA AND FACTOR ANALYSIS The equations presented in Chapter 2 were often complex owing to the numerous elements and subscripts involved. Matrix algebra is a simpler procedure for handling such equations. It is the approach to factor analysis. Section 3.1 (Matrix Definitions and Notation) contains the basic notation system of matrix algebra. Matrix operations that parallel the usual algebraic addition, subtraction, multiplication, and division are given in Section 3.2 (Matrix Operations). 1 An understanding of the basics of matrix algebra permits a simpler presentation of the material in Chapter 2. In Section 3.3, the component model is described in matrix algebra; the common factor model and uncorrelated models are developed in matrix algebra form in Sections 3.4 and 3.5. 3.1 Matrix definitions and notation A matrix is a rectangular array of numbers or functions. A capital letter is used to signify the entire array, or table, and brackets are usually placed around the numbers to indicate that they are to be considered as a unit. When it is necessary to refer to individual elements within the matrix, lower-case letters are used. Example The following is a simple matrix of three rows and two columns. Its elements are designated as follows: A number of matrices appear in developing factor-analytic concepts. The capital letters that are used to signify frequently used matrices are as follows: (1) X, a data matrix in deviation score form or Z if it is in standard score form; (2) F, the factor score matrix in standard score form; (3) C, the variance–covariance matrix where variances form the diagonal elements and covariances appear in the off-diagonals; (4) R, a correlation matrix among a set of variables or factors; (5) P, a factor pattern matrix giving the weights by which the factor scores are to be multiplied to give the standardized variable scores; and (6) S, the factor structure matrix that contains correlations of each variable with each factor...
