By Stefan Waner, Steven R. Costenoble
Take calculus into the true global with utilized CALCULUS. Authors Waner and Costenoble make utilized calculus effortless to appreciate and suitable on your pursuits. And, this textbook interfaces together with your graphing calculator and your house spreadsheet software. Plus it comes with AppliedCalculusNOW. After an easy pre-test, the AppliedCalculusNOW on-line studying process customizes all of the workouts and sophistication info round your personal wishes. This variation additionally comes with own educate with SMARTHINKING, which supplies you entry to one-on-one, on-line tutoring support with a professional within the topic. And it offers a digital learn workforce, too?interact with the train and different scholars utilizing two-way audio, an interactive whiteboard for discussing the matter, and rapid messaging.
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Additional info for Finite mathematics
Here, a = 1 and d = −1. Because the only factors of ±1 are ±1, the only possible rational solutions are x = ±1. By substitution, we see that x = 1 is a solution. Thus (x − 1) is a factor. Dividing by (x − 1) yields the quotient (x2 + 1). Thus, x 3 − x 2 + x − 1 = (x − 1)(x 2 + 1) = 0 so that either x − 1 = 0 or x 2 + 1 = 0. Because the discriminant of the quadratic x2 + 1 is negative, we don’t get any real solutions from x2 + 1 = 0, so the only real solution is x = 1. There may be irrational solutions, however; for example x 3 − 2 = 0 has the single solution x = † Alternatively, use “synthetic division,” a shortcut that would take us too far afield to describe.
In fact, the cube root of a always has the same sign as a. Higher roots are defined similarly. The fourth root of the nonnegative√number a is defined as the nonnegative number whose fourth power is a, and written 4 a. The fifth root of any number a is the number whose fifth power is a, and so on. Note We cannot take an even-numbered root of a negative number, but we can take an odd-numbered root of any number. Even roots are always positive, whereas odd roots have the same sign as the number we start with.
X 2 + 4x y + 4y 2 46. 4y 2 − 4x y + x 2 47. x 4 − 5x 2 + 4 48. y 4 + 2y 2 − 3 2 18. (3x + 1)(2x 2 − x + 1) 19. (x 2 − 2x + 1) 2 20. (x + y − x y) 2 21. ( y 3 + 2y 2 + y)( y 2 + 2y − 1) 2 22. (x 3 − 2x 2 + 4)(3x 2 − x + 2) In Exercises 23–30, factor each expression and simplify as much as possible. 23. (x + 1)(x + 2) + (x + 1)(x + 3) 24. 4 Rational Expressions Rational Expression P , where P and Q are Q simpler expressions (usually polynomials) and the denominator Q is not zero. A rational expression is an algebraic expression of the form quick Examples x 2 − 3x x x + x1 + 1 2.