AlgebraReview - Algebra Review 0.1 Monomial Factors Factor...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Algebra Review 0.1 Monomial Factors Factor as indicated: (a) 3x4 x (c) e (e) (g) 4x3 x xe x 2 x2 x2 2x2e x x e (d) x x 2 6x2 1 1 (f) sin x 4x x2 2 2 x tan x 2x x 1 x 1 sin x 1 2x 4x3 6x3 (b) 2 x 2x2 Solution: (a) 3x4 6x3 (b) 2 x (c) e x xe (d) x 1 2 (e) x 2 x2 3x2 2 x 2 x1 2x2e x x x 1 2 6x2 4x 3x x x e 1 1 x 1 x 2x2 2x x2 1 1 12x (g) tan x 0.2 1 cos x sec x 1 1 2x x 2 1 2x2 sin x cos x sin x sin x 1 sin x 1 (f) sin x 4x Binomial Factors Factor as indicated: (a) x 2 1 x (b) 3 x2 4 (c) x2 x 1 (d) x 3 (e) 2x 3 1 x2 1 1 x 6 1 x2 4 x2 x2 x 32 x1 2 3 x2 4 1 1 x2 2x 2 2 2 3 2x 1 1 x 2 52 x 2 x 12 2x 2 x 1 3 32 2 x 12 Solution: (a) x 1 2 x 1 4 1 6 x2 1x 1 x2 4 2 1x x 1 1 3 x2 4 x2 1 3 (b) 3 x2 x x x2 x x2 3x2 9 4 2 x2 4 —CONTINUED— 1 2 Algebra Review —CONTINUED— x2 x2 1 2 x x2 1 1 x2 (c) 12 1 1 x2 3 (e) 2x x 2 32 1 2x 12 x 2 12 12 1 x2 x2 1 1 3 2x 1 x 2 52 x x 2 12 3 2 x 2x x (d) x 3 x2 x2 3 2 x 2 1 32 1 32 2x 2x 0.3 3 x 2x 2 7 x 12 x 2x x 12 3x 1 1 Factoring Quadratic Expressions Factor as indicated: (a) x2 2 (c) x2 (e) 2x2 5x (b) x2 6 3x (d) x2 5x 6 e2x 2 e 5x (f) 7x 2 (g) x 4 3 12 (h) 1 9 2x 2 sin2 x Solution: (a) x2 3x 2 x 2x (b) x2 1 9 x 3x 3 (c) x2 5x 6 x 6x 1 (d) x2 5x 6 x 2x 3 (e) 2x2 5x (f) e2x 2 (g) x4 3 7x2 0.4 1x 2x ex e 12 e x2 3 3 x2 2 sin x (h) 1 2x 1 x2 sin x 1 4 x2 3x 2x 2 sin x Cancellation Reduce each expression to lowest terms: (a) (d) 3x 9 (b) 6x x1 2 13 x x1 (e) 6 Solution: (a) (b) 3x 9 3x 3 3 2x 6x x2 x1 2 x1 2 x3 2 12 x —CONTINUED— x 3 2x x3 2 x2 x1 2 x (c) 1 x x 1 1 32 (f) x 1 3 x 2 x 1 3x 1 4 sin x cos x 2 sin x 2 1 2 S ection 0.5 —CONTINUED— (c) x 3 1 x 2 3x x 2 1 x 2 1 1 x x2 x x1 2 x1 x1 x (e) 3 x1 6 x2 6 x1 6 1 x x x1 x1 x 2 3 x1 11 x 1 sin x cos x 2 sin x 1 6 x 1 1 x 1 sin2 x 2 sin x cos x 2 sin x sin2 x cos2 x 2 sin x 1 0.5 1 cos2 x 2 sin x cos x 2 sin x cos x 2 sin x cos x Quadratic Formula Equation (a) 3 2 1 1 (f) 2 6 6 32 1 1x 14 1 x (d) x 4 x2 Solve for 4x 1 0 x x 3 0 x (b) 2x2 (c) cos2 x 3 cos x 2 (d) x2 xy 1 y2 (e) x4 4x2 2 16 2 cos x 0 4± 0 0 x x2 Solution: (a) x 1± (b) x x 4± 4 4 1 3±1 2 8 2 2 2 y± 1 y2 41 or cos x y2 y± 2 4± 16 2 4 2 2 y2 4y2 4 2 y± (e) x2 5 3 2 6 4 1 or x 9 2± 1±5 4 24 3± cos x 4±2 5 2 20 2 4 (c) cos x (d) x 4 8 4± 8 2 5y2 2 4 4±2 2 2 2± 2 Quadratic Formula 3 4 Algebra Review 0.6 Synthetic Division Using synthetic division to factor as indicated: 4x2 2x 1 (c) x4 3x3 x2 x 4x2 2x 4 1 2 3 3 1 4x2 2x 1 (c) x4 3x3 x2 x 5x 1 2 0 2 5 2 7 7 2 2 7 0 0 x3 3x2 1 1 1 5x (d) 4x4 1 1 (b) 2x3 (b) 2x3 (a) x3 x 1 2 x 2 7 x 1 1 2x 1 Solution: (a) x3 1 2 1 x2 x 3x (d) 4x4 2 1 2 3 2 1 2 1 2 2 2 1 1 1 1 3x3 x2 x 2 x3 2 7 3x2 1 2x2 x 2x x2 x 1 0 2 3 1 0 2 1 1 2 4 2 0 0.7 3x2 x 1 1 4x3 2 2x2 2x 4x4 7 1 4 0 x 5x 4 1 x4 2x3 1 7 1 2x3 x2 4x 2x Special Products Factor completely (into linear or irreducible quadratic factors): (a) x3 27 (b) x3 3x2 3x (d) x4 25 (e) x4 8x3 24x2 (c) x3 1 32x 6x2 16 Solution: (a) x3 27 x 3 x2 3x (b) x3 3x2 (c) x3 6x2 12x 8 (d) x4 25 x2 5 x2 (e) x4 8x3 24x2 0.8 3x 1 x 9 1 x3 3 3 2 x2 23 x 2 3 x2 5x 5x 5 x4 5 32x 3 22 x 4 2 x3 6 22 x2 4 23 x 16 24 x 2 4 Factoring by Grouping Factor completely (into linear or irreducible quadratic factors): (a) x3 4x2 (c) 5 cos2 x 2x (b) x3 8 5 sin2 x sin x (d) cos2 x cos x Solution: (a) x3 4x2 8 x2 x 4 2x x2 2x 2x 4 x —CONTINUED— 2x 2x2 4 2x 4 3x 4 cos x 6 4 tan2 x 12x 8 2 1 S ection 0.10 Rationalizing —CONTINUED— (b) x3 2x2 cos2 x2 x 6 3x 3x 2 2 5 (d) cos2 x cos x tan2 x 4 sin x cos x sin x 5 cos x cos x 2 cos x 0.9 4 cos x sin x sin2 x cos x x 5 cos2 x 5 cos x (c) 5 sin2 x 2 x2 3x 2 sin x cos x sin x cos x sin x sin x 1 tan2 x 2 tan x cos x 2 tan x Simplifying Rewrite each of the following in simplest form: (a) x 1x 3 x x (c) x2 x2 (e) x2 2 6 4 (d) x2 1 x 1 1 1 1 2 x x2 x2 1 x2 1 1 1 x2 1 1 x2 1 (b) 1 x 2x 2 1 x2 5x 4x 1 x2 1 Solution: (a) x 1x (e) 1 x2 x2 x2 x2 5x 4x x x x2 1 x2 1 x x 1 1 2 2x 1 x2 1 6 4 1 x 1 2x 2 2 x2 1 2x 1 x 1 x x2 x2 1 2 0.10 11 1 x2 1 32 x2 x2 4 x2 1 1 1 4 1 3 2 x2 x2 1 x 1 1 1 2x 1 1 x2 21 1 21 x2 x2 x x 3 2 x2 1 3 x 1 1 (b) (d) x x x2 (c) 3 x2 x2 1 1 x2 1 x2 21 1 1 2 x2 2x2 1 x2 x2 Rationalizing Remove the sum or difference from the denominator by multiplying the numerator and denominator by the conjugate of the denominator. (a) 1 1 cos x (b) Solution: 1 1 cos x 1 1 cos x 1 1 (a) cos x cos2 x —CONTINUED— 1 1 1 cos x cos x cos x sin2 x 1 x x2 1 (c) x 2 x2 1 5 6 Algebra Review —CONTINUED— (b) x x2 1 1 1 1 x2 x2 1 1 x2 x1 1 2 x x2 x2 x1 (c) 2 x2 x 0.11 1 x x b 1 a 2 a 2 b a2 1 b 2 a x b2 b a x 1 a 1 b a x a b a b 1x 1 a b 1 x 2x 2x 1 x2 1 3 5x a2 3 bx ax a 1 bx a x b2 Don’t forget middle term when squaring binomials. b Use definition for adding fractions. ab 1 x x ab 1 x x2 bx a a ax a x6 Be careful when using a slash to denote division. Multiply exponents when an exponential form is raised to a power. Exponents have priority over coefficients. 1 x2 Don’t shift term-by-term from denominator to numerator. x3 5x Leave as a 3 2 2 x3 Leave as Multiply by reciprocal of the denominator. Use definition for multiplying fractions. 2 3 1 2 occurs twice as a factor. Don’t add denominators when adding fractions. b a 5x x Change all signs when distribution negative through parentheses. a x 1 3 5x x2 x b 2ab 1 b 2x3 2 x 1 ab 4 x a x2 x x3 a 1 b 2 1 b 1x 2 x5 1 a 1 1 x2 x 1 Comments a2 1 3x 2 a x a 2 b Leave as 1 x 3 a 2 b x a b 1 3x x2 x 1 a 2 bx a 2x3 a a 1 3 b 1 ab 2 x x2 x2 2x x2 Correct form x a 1 1 1 Algebraic Errors to Avoid Error a x2 x2 1 1 Radicals apply to every factor inside radical. x2 a2 Don’t apply radicals term-by-term. 1 b x a Cancel common factor, not common terms. 1 x Factor before canceling. ...
View Full Document

This note was uploaded on 10/05/2009 for the course MATH calculus 1 taught by Professor Man during the Spring '09 term at Lakehead.

Ask a homework question - tutors are online