Dawson College Mathematics Functions and Trigonometry PDF

DAWSON C O L L E G E Mathematics Department Functions and Trigonometry f (x) © CONTENT CHARTS, EXAMPLES AND EXERCISES By (2nd Edition) (Dec. 2022) George McArthur Contents...

DAWSON C O L L E G E Mathematics Department Functions and Trigonometry f (x) © CONTENT CHARTS, EXAMPLES AND EXERCISES By (2nd Edition) (Dec. 2022) George McArthur Contents Basic Algebra 1 1 Integer Exponents In Algebra 1 2 Polynomials I 4 3 Polynomials II 8 4 Factoring 11 5 Rational Expressions (Fractions) 17 6 Roots and Radicals 24 7 Solving Linear Equations 30 8 Solving Linear Systems 34 9 Solving Quadratic Equations by Factoring 39 10 Solving Quadratic Equations (Continued) 46 11 Solving Equations Containing Fractions 53 12 Solving Radical Equations 58 13 Solving Linear Inequalities 62 Functions 66 14 The Rectangular Coordinate System 66 15 Introduction to Functions 71 16 Linear Functions 79 17 The Slope and Equation of a Line 84 18 Quadratic Functions 94 19 Solving Systems of Equations 103 20 Piece-wise Defined Functions 107 21 Operations on Functions and Composite Functions 112 22 Inverse Functions 118 23 Exponential and Logarithmic Functions 124 i 24 Logarithmic Functions 134 Trigonometry 144 25 Angles in Trigonometry 144 26 The Trigonometric Functions 148 27 Solving Right Triangles 152 28 Solving Oblique Triangles 160 29 The Special Angles and Reference Angles 173 30 Radian Measure in Trigonometry 177 31 Graphs of the Sine and Cosine Functions 183 32 Trigonometric Identities 189 33 Solving Trigonometric Equations 194 34 The Inverse Trigonometric Equations 199 Geometry 202 35 Introduction to Vectors 202 36 Components of Vectors 207 37 The Dot Product 211 38 Similar Figures 214 ii  Basic Algebra 1 Integer Exponents In Algebra THE DEFINITION OF AN EXPONENTIAL EXPRESSION an is an exponential expression where a is the BASE and n is the EXPONENT (or POWER ) such that: an = a| · a · {z a ·... · a} n factors 1 a0 = 1, a 6= 0 a n = , a 6= 0 an NOTE: an is read “a to the nth power”, a1 = a, a2 is read “a squared,” and a3 is read “a cubed”. THE RULES OF EXPONENTS PRODUCT am · an = am+n POWER (am )n = amn am QUOTIENT = am n , a 6= 0 an PROPERTIES 1 (ab)n = an bn a m bn 3 = ⇣ a ⌘n an b n am 2 = n ⇣ a ⌘ n ✓ b ◆n b b 4 = b a 1  INTEGER EXPONENTS IN ALGEBRA - EXAMPLES Simplify, expressing the answers with positive exponents only. Example 1: a2 · a3 a2+3 a5 Product Rule = 2·3 = 6 (a2 )3 a a Power Rule = a5 6 Quotient Rule 1 =a 1 = Definition a Example 2: ✓ ◆2 2a2 (2a2 )2 = Property 2 3b3 (3b3 )2 22 (a2 )2 = Property 1 32 (b3 )2 4a4 = Power Rule 9b6 Example 3: (3x2 ) 2 (2y3 )3 = Property 3 (2y3 ) 3 (3x2 )2 23 (y3 )3 = Property 1 32 (x2 )2 8y9 = Power Rule 9x4 Example 4: ✓ ◆ 4 ✓ ◆4 x 2 2y3 = Property 4 2y3 x 2 (2y3 )4 = Property 2 (x 2 )4 16y12 = Power Rule x 8 = 16y12 x8 Definition 2 1.1 Integer Exponents In Algebra – Exercises Simplify, expressing the answers with positive exponents only: 5 ✓ ◆ 2 1. a2 · a3 14. 2x2 y3 4x 1 y 40 23. 3 2 2 x4 y 10 2. a2 15. 5a2 b3 2a 2 b 4 ✓ ◆ 2 a5 3 4x 3 3. 2 6x15 2x0 2 24. a 16. 3 3x 6 ( 2x4 )3 4. x2 · x3 · x4 2xy 3 2 ✓ 2 ◆ 4 25. 3 x7 x9 5x (3x 2 y4 ) 5. 17. (x3 )2 y 2 3 (xy) (xy)5 6. 2x3 7x2 x 0 y 2 z3 26. 4 18. (xy) 1 2 (xy 1 z 3 ) " #0 7. 6x2 2x4 3 ✓ ◆ 2 a55 b23 3a2 b 2 27. 9x5 19. a 16 8. ab2 3x2 4 a 2 b3 9. 2x3 4 a 3 ⇣ a ⌘3 28. 20. (a 3 b2 ) 2 (ab) 4 3 · x5 · x 4 b 3 b 10. x ✓ 2 ◆2 4 3 x x11 x2 11. ( 3x) ( 3x) 21. 29. 2y 1 (x3 )3 (x2 ) 6 1 12. 2x2 4x3 0 3 x 4 y 2 z2 2xy 3 3 5 2 22. 30. 13. x2 x3 x4 (x4 y 2 z2 ) 2 (xy 1 ) 1 1.2 Integer Exponents In Algebra – Answers 1. a5 8. 3x3 14. 32x10 y15 20. 1 27y18 25. 10 4x8 2. a6 9. 16x12 15. y2 b 21. 26. x6 y6 4x4 3. a3 1 16. 3x3 10. x 8 z4 27. 1 x2 1 22. 4. x9 17. y4 a6 11. 81x4 625x8 y8 28. 5. x10 x x10 y60 b4 1 18. 3 23. 12. y 256 29. x22 6. 14x5 2x b8 9 y8 7. 24x10 13. x13 19. 24. 30. 9a2 x6 8x2 3  2 Polynomials I THE DEFINITION OF A POLYNOMIAL A Polynomial in x is a sum of terms that may be denoted, in descending powers of x, as follows: an xn + an 1 xn 1 +···+a 1 x + a0 The DEGREE of the polynomial is the non-negative integer n. The COEFFICIENTS of the polynomial are the real numbers an , an 1 , · · · , a1 , a0 NOTE: A polynomial with 1 term is called a MONOMIAL. A polynomial with 2 term is called a BINOMIAL. A polynomial with 3 term is called a TRINOMIAL. Example: 3x2 5x + 1 is a trinomial of degree 2 with coefficients 3, 5, and 1. ADDING AND SUBTRACTING POLYNOMIALS Combine like terms NOTE: Like terms have the same variable(s) and exponent(s). MULTIPLYING POLYNOMIALS a(c + d) = ac + ad (a + b)(c + d) = a(c + d) + b(c + d) SPECIAL BINOMIAL PRODUCTS SQUARES OF A BINOMIAL (x + y)2 = x2 + 2xy + y2 and (x y)2 = x2 2xy + y2 NOTE: These products are perfect square trinomials. PRODUCT OF A BINOMIAL SUM AND DIFFERENCE (x + y)(x y) = x2 y2 NOTE: This product is a difference of squares. (a + b)(c + d + e) = a(c + d + e) + b(c + d + e) 4  POLYNOMIALS I - EXAMPLES Example 1: Add or subtract as indicated: (5x2 x + 3) (2x2 7x + 6) = 5x2 x+3 2x2 + 7x 6 = 3x2 + 6x 3 Example 2: Multiply and simplify: a(b + c) = ab + ac 3x2 (x 2) = 3x3 6x2 (2x + 5) = ( 1)(2x + 5) = 2x 5 Example 3: Multiply and simplify: (x + 5)(x 3) = x(x 3) + 5(x 3) 2 =x 3x + 5x 15 2 = x + 2x 15 Multiplying two Binomials by the FOIL METHOD First Last (x + 5)(x 3) = x2 3x + 5x 15 = x2 + 2x 15 Inner F O I L Outer F: First terms product O: Outer terms product I: Inner terms product L: Last terms product Example 4: Find the special products and simplify: (a + b)2 = a2 + 2ab + b2 (a b)2 = a2 2ab + b2 (a + b)(a b) = a2 b2 (x + 3)2 = x2 + 2 · x · 3 + 32 = x2 + 6x + 9 (2x 5)2 = (2x)2 2 · 2x · 5 + 52 = 4x2 20x + 25 2 2 2 (3x + 4)(3x 4) = (3x) 4 = 9x 16 Example 5: Multiply and simplify: (a + b)(a2 ab + b2 ) = a(a2 ab + b2 ) + b(a2 ab + b2 ) = a3 a2 b + ab2 + a2 b ab2 + b3 ) = a3 + b3 , a sum of cubes. Similarly, (a b)(a2 + ab + b2 ) = a3 b3 , a difference of cubes. 5 2.1 Polynomials I – Exercises 1. For each polynomial, give its degree and name (if applicable): (a) 4x5 5,monome (c) 7x2 4x + 6 2 trinome (e) 25x4 16 4 binome (b) 3x2 + 5 2 binome (d) x3 + 2x2 + 3x 4 5 poly 2. Add or subtract as indicated: (a) 3x2 + 5x2 (e) 5x2 + 4x 6 x2 2x + 1 (b) (5x + 4) + (2x 3) (f) 25x3 + 14x 8x2 + 9x + 1 (c) 7x2 6x + 5 + x2 + 4x 2 (g) 7x4 + 3x2 + 2x 18x4 5x2 + x (d) (6x + 5) (3x + 1) (h) 10x4 + 11x2 10x + 6 + 11x2 + 10x 15 (i) 9x2 + 2x 1 11x2 + 5x 8 + 2x2 + 4x 7 (j) 2x2 + x 5 3x2 + 2x 2 + x2 + x + 3 3. Multiply and simplify: (a) x (x + 2) (f) x2 3x2 x+6 (b) 3x (5x 8) (g) 10 5x2 + 7x 4 (c) x3 (x + 12) (h) 4x2 3x3 12x2 6 (d) 4x2 (x 7) (i) 5x4 x3 2x2 3x 4 (e) 2x3 x2 5x (j) 9x5 3x6 2x4 + 8x2 4. Multiply and simplify (you may use FOIL): (a) (x + 4) (x + 7) (f) (7x + 2) (3x 4) (b) (x 5) (x + 2) (g) (4x 1) (6x 5) (c) (2x 1) (x + 3) (h) 2x2 7 5x2 + 3 (d) (3x + 4) (4x + 3) (i) x 2x2 5 x2 + 3 (e) (2x + 5) (5x 7) (j) 3x3 (x 5) (2x + 3) 5. Find the special products and simplify: (a) (x + 1)2 (f) (x + 5) (x 5) 2 (b) (x 1) (g) (4x + 3) (4x 3) (c) (2x + 5)2 (h) 5x2 + 4 5x2 4 2 2 (d) (4x 3) (i) x5 3x2 1 (e) (x + 1) (x 1) (j) 7x3 (3x + 2) (3x 2) 6. Multiply and simplify: (a) (x 3) x2 + 2x 1 (d) (2x 7) x2 6x + 1 (b) (x + 1) x2 5x + 3 (e) (x 1) x2 + x + 1 (c) (x + 3) 2x2 4x + 3 (f) (2x + 3) 4x2 6x + 9 6  (g) 10 (4x 1) 16x2 + 4x + 1 (i) (4x + 3) 5x3 4x2 + x 5 (h) 3x3 2x2 + 5x x3 + 2x + 1 (j) x2 + 2x + 1 3x2 6x 1 7. Simplify: (a) (x + y)3 (f) 2 (x 3)2 + 3 (x + 3) (x 3) 3 (b) (x y) (g) 5 x2 + y2 (x + y) (x y) (c) (x + 1)3 (h) 2x3 x2 2x + 3 (2x 1) 3 3 2 (d) (2x 3) (i) (x 1) (x + 1) + 4 (x + 1) (x 1) (e) 4x2 (x + 1) 2x x2 + 2x (j) (x + 2) x2 2x + 4 (x 2) x2 + 2x + 4 2.2 Polynomials I – Answers 1. (a) 5, monomial 3. (a) x2 + 2x (f) 21x2 22x 8 (b) 2, binomial (b) 15x2 24x (g) 24x2 26x + 5 (c) 2, trinomial (c) x4 + 12x3 (h) 10x4 29x2 21 (d) 3, polynomial (d) 4x3 28x2 (i) 2x5 + x3 15x (e) 4, binomial (e) 2x5 + 10x4 (j) 6x5 21x4 45x3 2. (a) 8x2 (f) 3x4 x3 + 6x2 5. (a) x2 + 2x + 1 (b) 7x + 1 (g) 50x2 + 70x 40 (b) x2 2x + 1 (c) 8x2 2x + 3 (h) 12x5 + 48x4 + 24x2 (c) 4x2 + 20x + 25 (d) 3x + 4 (i) 5x7 10x6 15x5 20x4 (d) 16x2 24x + 9 (e) 4x2 + 6x 7 (j) 27x11 + 18x9 72x7 (e) x2 1 (f) 25x3 8x2 + 5x 1 4. (a) x2 + 11x + 28 (f) x2 25 (g) 11x4 + 8x2 + x (b) x2 3x 10 (g) 16x2 9 (h) 10x4 9 (c) 2x2 + 5x 3 (h) 25x4 16 (i) x (d) 12x2 + 25x + 12 (i) 9x9 6x7 + x5 (j) 0 (e) 10x2 + 11x 35 (j) 63x5 28x3 6. (a) x3 x2 7x + 3 7. (a) x3 + 3x2 y + 3xy2 + y3 (b) x3 4x2 2x + 3 (b) x3 3x2 y + 3xy2 y3 (c) 2x3 + 2x2 9x + 9 (c) x3 + 3x2 + 3x + 1 (d) 2x3 19x2 + 44x 7 (d) 8x3 36x2 + 54x 27 (e) x3 1 (e) 2x3 (f) 8x3 + 27 (f) 5x2 12x 9 (g) 640x3 10 (g) 5x4 5y4 (h) 6x8 + 15x7 + 12x6 + 36x5 + 15x4 (h) 4x6 + 6x5 16x4 + 6x3 (i) 20x4 x3 8x2 17x 15 (i) x3 + x 6 (j) 3x4 10x2 8x 1 (j) x6 64 7  3 Polynomials II DIVIDING POLYNOMIALS Illustrative Example 3x2 + 4x 3 Divide by Long Division as follows: x+2 3x2 2x Step 1 3 x x 3x 2 x+2 3x2 + 4x 3 3x (x + 2) (3x2 + 6x) 2 subtract 2x 3 ( 2x 4) 2 (x + 2) 4 1 subtract Dividend Divisor Quotient Remainder z }| { z }| { z }| { z}|{ Check: 3x2 + 4x 3 = (x + 2) · (3x 2) + 1 ? = 3x2 2x + 6x 4+1 ? = 3x2 + 4x 3 True 3x2 + 4x 3 1 Hence, = 3x 2+ x+2 x+2 Note: 1 In long division, both the dividend and the divisor must be written in descending powers of x. 2 Long division is complete when the degree of the remainder is less than the degree of the divisor. 3 In either the dividend or the divisor, any missing terms in descending powers of x, must be entered with a 0 coefficient to keep like terms in the same column. 8  POLYNOMIALS II - EXAMPLES 3x3 5x + 2 Example 1: Divide x 1 3x2 + 3x 2 Quotient x 1 3x3 + 0x2 5x + 2 (3x3 3x2 ) 3x2 5x (3x2 3x) 2x + 2 ( 2x + 2) 0 Remainder 2x4 x3 + 3x2 + x + 2 Example 2: Divide x2 + 1 2x2 x+1 Quotient x2 + 0x + 1 2x4 x3 + 3x2 + x + 2 (2x4 + 0x3 + 2x2 ) x3 + x2 + x ( x3 0x2 x) x2 + 2x + 2 (x2 + 0x + 1) 2x + 1 Remainder 6x3 11x2 + 8x 7 Example 3: Divide 2x 1 3x2 4x + 2 Quotient 2x 1 6x3 11x2 + 8x 7 (6x3 3x2 ) 8x2 + 8x ( 8x2 + 4x) 4x 7 (4x 2) 5 Remainder 9 3.1 Polynomials II – Exercises Divide by long division to find the quotient and remainder: 4x2 + 7x + 3 3x x2 + 2x3 + 2 x 4 + 3x3 1. 11. 21. x+1 2x + 1 4+x x2 + 7x 2 4x3 + 8x2 x + 6 x4 6x2 + 5x + 4 2. 12. 22. x+5 2x 1 x 2 x2 3x 20 6x3 3x2 + 14x 7 x4 + 4x3 5x2 12x + 6 3. 13. 23. x 4 2x 1 x2 3 x2 x 3 6x3 5x2 + 2x + 1 x4 + 2x3 + 2x2 x 1 4. 14. 24. x 2 4+x x2 + 1 6x2 + x 2 x4 4x3 + 6x2 4x + 1 x4 5x2 + 4 5. 15. 25. 2x 1 x 1 x2 1 4x + 3x2 1 x4 2x3 + 5x2 4x + 3 x5 2x3 + 5x + 1 6. 16. 26. x 1 x+1 x2 x + 2 x3 2x2 5x + 10 x3 + 2x2 4 x3 x2 + x + 3 7. 17. 27. x 1 x 3 x2 2x + 3 3x3 + 5x2 6x + 18 2x3 + x 18 3x4 10x2 2x + 2 8. 18. 28. x+3 x 2 x2 + 2x + 1 5x3 11x2 + 8x 12 5x3 + x2 + 4 6x3 + 7x2 18x + 15 9. 19. 29. x 2 x+1 2x2 + 3x + 5 5x3 + 12x2 + x 3 2x3 + 5x2 1 x5 2x3 3x2 + 9 10. 20. 30. x+2 x 2 x2 2 36x4 + 72x3 121x2 142x + 120 12x + 11x + 3x + 10x3 9x2 + 3x 6 5 4 6 31. 32. 6x2 + 11x 10 4x4 + 5x3 3 3.2 Polynomials II – Answers 1. 4x + 3 and 0 12. 2x2 + 5x + 2 and 8 23. x2 + 4x 2 and 0 2. x + 2 and 12 13. 3x2 + 7 and 0 24. x2 + 2x + 1 and 3x 2 3. x + 1 and 16 14. 6x2 + 19x + 78 and 313 25. x2 4 and 0 4. x + 1 and 1 15. x3 3x2 + 3x 1 and 0 26. x3 + x2 3x 5 and 6x + 11 5. 3x + 2 and 0 16. x3 3x2 + 8x 12 and 15 6. 3x + 7 and 6 17. x2 + 5x + 15 and 41 27. x + 1 and 0 7. x2 x 6 and 4 18. 2x2 + 4x + 9 and 0 28. 3x2 6x 1 and 6x + 3 8. 3x2 4x + 6 and 0 19. 5x2 4x + 4 and 0 29. 3x 1 and 30x + 20 9. 5x2 x + 6 and 0 20. 2x2 + 9x + 18 and 35 30. x3 3 and 3 10. 5x2 + 2x 3 and 3 21. 3x2 12x + 49 and 200 31. 6x2 + x 12 and 0 11. x2 x + 2 and 0 22. x3 + 2x2 2x + 1 and 6 32. 3x2 x + 2 and 0 10  4 Factoring GREATEST COMMON FACTOR ax + ab = a(x + b) TRINOMIALS (with coefficient of x2 = 1) x2 + (a + b)x + ab = (x + a)(x + b) TRINOMIALS (with coefficient of x2 6= 1) acx2 + (ad + bc)x + bd = (ax + b)(cx + d) SPECIAL FACTORIZATIONS PERFECT SQUARE TRINOMIALS x2 + 2xy + y2 = (x + y)2 x2 2xy + y2 = (x y)2 DIFFERENCE OF SQUARES x2 y2 = (x + y)(x y) DIFFERENCE AND SUM OF CUBES x3 y3 = (x y)(x2 + xy + y2 ) x3 + y3 = (x + y)(x2 xy + y2 ) NOTE: A trinomial ax2 + bx + c can be factored over the integers only if b2 4ac = 0, 1, 4, 9, 16, · · ·. Also, if b2 4ac = 0, it is a perfect square trinomial. 11  FACTORING - EXAMPLES Factor each polynomial: Greatest Common Factor (GCF): Recall: The GCF is the largest factor that all terms have in common. 5x2 10x = (5x)(x) (5x)(2) = 5x(x 2) Check: multiply 5x(x 2) = 5x2 10x X 12x5 y5 18x3 y3 3x2 y3 = (3x2 y3 )(4x3 y2 ) (3x2 y3 )(6x) (3x2 y3 )(1) = (3x2 y3 )(4x3 y2 6x 1) Check: multiply (3x2 y3 )(4x3 y2 6x 1) = 12x5 y5 18x3 y3 3x2 y3 X Common Factors in Factoring by Grouping: 2x2 2x + 3x + 3 = (2x2 2x) + (3x 3) , grouping = 2x(x 1) + 3(x 1) , common factors = (x 1)(2x + 3) , common binomial factor Check: multiply (x 1)(2x + 3) = 2x2 + 3x 2x 3 = 2x2 2x + 3x 3X Trinomials (with coefficient of x2 = 1): x2 + 2x 8 = (x + a)(x + b) where ab = 8 and (a + b) = 2 = (x + 4)(x 2) Check: multiply (x + 4)(x 2) = x2 + 2x 8X Trinomials (with coefficient of x2 6= 1): We factor this form by grouping as follows: Factor 2x2 + 3x 9, consider the product (2x)( 9) = 18x2 = (6x)( 3x) such that 6x 3x = 3x and write 2x2 + 3x 9 = 2x2 + 6x 3x 9 = 2x(x + 3) 3(x + 3) = (x + 3)(2x 3) , factor by grouping Check: multiply (x + 3)(2x 3) = 2x2 + 3x 9 Note: If necessary, we can factor the other trinomial forms by grouping also. 12  FACTORING - EXAMPLES Note: Recall that all factoring results can be checked by multiplying. Perfect Square Trinomials: x2 + 2x + 1 = x2 + 2 · x · 1 + 12 , hence = (x + 1)2 4x2 12x + 9 = (2x)2 + 2 · (2x) · 3 + 32 , hence = (2x 3)2 Difference of Squares: x2 4 = x2 22 , hence 9x2 16 = (3x)2 42 , hence = (x + 2)(x 2) = (3x + 4)(3x 4) Difference and Sum of Cubes: x3 8 = x3 23 , hence 8x3 + 27 = (2x)3 + 33 , hence = (x 2)(x2 + 2x + 22 ) = (2x + 3)((2x)2 (2x)(3) + 32 ) = (x 2)(x2 + 2x + 4) = (2x + 3)(4x2 6x + 9) Greatest Common Factor as First Step: 2x3 14x2 + 36x = 2x(x2 7x + 18) , greatest common factor = 2x(x 9)(x + 2) 300x3 y2 3xy4 = (3xy2 )(100x2 y2 ) , greatest common factor = (3xy2 )((10x)2 y2 ) , difference of squares = (3xy2 )(10x + y)(10x y) Note: Not all trinomials are factorable over integers. Consider 2x2 + 3x 1, then b2 4ac = 32 4(2)( 1) = 9 + 8 = 17, hence it is not factorable since 17 is not a perfect square integer. 13 4.1 Factoring – Exercises 1. Factor out the greatest common factor: (a) 10x + 20 (f) 15x5 18x4 + 21x3 48x2 (b) 18x3 9x (g) 56x5 y4 + 21x3 y2 35x2 y3 (c) 28x5 + 14x4 21x3 (h) x (x + 5) + 4 (x + 5) (d) 50x2 y2 10xy2 (i) 2x (x 1) 3 (x 1) (e) 2x3 y 6x2 y2 + 14xy3 (j) 7x2 (x + 1)2 + 8x (x + 1)2 2. Factor by grouping: (a) x2 + 3x + 2x + 6 (e) 3x2 9x 8x + 24 (i) 3xy y2 + 3x y (b) x2 5x + 4x 20 (f) 5x2 10x x + 2 (j) 3x3 + 3x2 2x 2 (c) x2 + 7x 2x 14 (g) 4x2 + 10x 6x 15 (d) 2x2 + 10x + 7x + 35 (h) x 1 + xy y 3. Factor each trinomial: (a) x2 + 4x + 3 (e) x2 + 5x 36 (i) x2 25x + 126 (b) x2 + 10x 11 (f) x2 2x 63 (j) x2 + 8x 105 (c) x2 + x 20 (g) x2 9x + 20 (d) x2 13x + 42 (h) x2 21x 100 4. Factor each trinomial: (a) 3x2 + 8x + 5 (f) 2x2 x 6 (k) 2x2 + 5x 18 (b) 2x2 + 5x 3 (g) 8x2 + 14x + 5 (l) 10x2 23x + 12 (c) 5x2 7x 6 (h) 7x2 27x 4 (m) 20x2 39x 11 (d) 6x2 + 7x 10 (i) 12x2 + 8x 15 (e) 4x2 12x + 5 (j) 21x2 + 25x 4 (n) 18x2 9x 5 5. Factor each perfect square trinomial: (a) x2 + 10x + 25 (e) 16x2 56x + 49 (i) 1 4x + 4x2 (b) x2 2x + 1 (f) 36x2 60x + 25 (j) 81x2 + 180x + 100 (c) x2 22x + 121 (g) 25x2 + 10x + 1 (d) 4x2 + 20x + 25 (h) 9x2 24x + 16 6. Factor each difference of squares: (a) x2 25 (e) 49x2 36y2 (i) 64x2 100 (b) x2 49 (f) 9x2 64y2 (j) x4 1 (c) 4x2 81 (g) 49 9x2 (d) 16x2 1 (h) 16x2 121y2 14  7. Factor each difference or sum of cubes: (a) x3 1 (e) 8x3 27y3 (i) 512x3 343 (b) x3 + 27 (f) 64x3 + 27 (j) 125x3 + 1000 (c) x3 125 (g) 27x3 125 (d) x3 + 64 (h) 8x3 + 729y3 8. Factor completely: (a) 27x3 15x (i) 12x5 + 12x3 4x4 4x2 (q) 25x3 + 65x2 30x (b) 50x3 100x2 10x2 + 20x (j) x6 64 (r) x3 3x2 4x + 12 (c) 2x6 + 8x5 42x4 (k) 54x4 + 2000x (d) 15x4 25x3 + 10x2 (l) x3 3x2 4x + 12 (s) 120x5 + 110x4 50x3 (e) 16x5 + 48x4 + 36x3 (m) (x 2)2 + 3 (x 2) (t) (x + 1)2 (x + 1) 6 (f) 3x3 24x2 + 48x (n) x2 (x 2) (x 2) 2 (u) x2 9 + 8x x2 9 (g) 10x3 270 (o) (5x + 7)2 16 (h) 16ax3 + 54ay3 (p) 7x4 + 7x3 140x2 (v) (x 1)3 8 9. Determine whether each trinomial is factorable over the integers or not. (a) x2 + 5x 3 (c) 3x2 15x + 16 (e) 2x2 + 5x 5 (b) x2 + 3x 88 (d) 5x2 + 13x 6 (f) 9x2 3x 2 4.2 Factoring – Answers 1. (a) 10 (x + 2) (h) (y + 1) (x 1) (b) 9x 2x2 1 (i) (y + 1) (3x y) (c) 7x3 4x2 + 2x 3 (j) (x + 1) 3x2 2 (d) 10xy2 (5x 1) 3. (a) (x + 3) (x + 1) (e) 2xy x2 3xy + 7y2 (b) (x + 11) (x 1) (f) 3x2 5x3 6x2 + 7x 16 (c) (x + 5) (x 4) (g) 7x2 y2 8x3 y2 + 3x 5y (d) (x 6) (x 7) (h) (x + 5) (x + 4) (e) (x + 9) (x 4) (f) (x + 7) (x 9) (i) (x 1) (2x 3) 2 (g) (x 4) (x 5) (j) x (x + 1) (7x + 8) (h) (x + 4) (x 25) 2. (a) (x + 3) (x + 2) (i) (x 7) (x 18) (b) (x + 4) (x 5) (j) (x + 15) (x 7) (c) (x + 7) (x 2) 4. (a) (3x + 5) (x + 1) (d) (x + 5) (2x + 7) (b) (x + 3) (2x 1) (e) (3x 8) (x 3) (c) (5x + 3) (x 2) (f) (5x 1) (x 2) (d) (x + 2) (6x 5) (g) (2x + 5) (2x 3) (e) (2x 1) (2x 5) 15  (f) (2x + 3) (x 2) (f) (4x + 3) 16x2 12x + 9 (g) (4x + 5) (2x + 1) (g) (3x 5) 9x2 + 15x + 25 (h) (7x + 1) (x 4) (h) (2x + 9y) 4x2 18xy + 81y2 (i) (2x + 3) (6x 5) (i) (8x 7) 64x2 + 56x + 49 (j) (3x + 4) (7x 1) (j) 125 (x + 2) x2 2x + 4 (k) (2x + 9) (x 2) 8. (a) 3x 9x2 5 (l) (5x 4) (2x 3) (b) 10x (5x 1) (x 2) (m) (4x + 1) (5x 11) (c) 2x4 (x + 7) (x 3) (n) (3x + 1) (6x 5) (d) 5x2 (3x 2) (x 1) 5. (a) (x + 5)2 2 (e) 4x3 (2x + 3) (b) (x 1)2 2 (f) 3x (x 4)2 (c) (x 11) 2 (g) 10 (x 3) x2 + 3x + 9 (d) (2x + 5) (h) 2a (2x + 3y) 4x2 6xy + 9y2 (e) (4x 7)2 (i) 4x2 x2 + 1 (3x 1) (f) (6x 5)2 (j) (x 2) (x + 2) x2 + 2x + 4 x2 2x + 4 (g) (5x + 1)2 (k) 2x (3x + 10) 9x2 30x + 100 (h) (3x 4)2 (l) (x 3) (x + 2) (x 2) (i) (2x 1)2 (j) (9x + 10)2 (m) (x + 1) (x 2) (n) (x 1) (x 2) (x + 1) 6. (a) (x 5) (x + 5) (o) (5x + 11) (5x + 3) (b) (x 7) (x + 7) (p) 7x2 (x + 5) (x 4) (c) (2x 9) (2x + 9) (q) 5x (x + 3) (5x 2) (d) (4x 1) (4x + 1) (r) (x 3) (x + 2) (x 2) (e) (7x 6y) (7x + 6y) (s) 10x3 (4x + 5) (3x 1) (f) (3x 8y) (3x + 8y) (t) (x + 3) (x 2) (g) (7 3x) (7 + 3x) (h) (4x 11y) (4x + 11y) (u) (x 1) (x 3) (x + 3) (x + 9) (i) 4 (4x 5) (4x + 5) (v) (x 3) x2 + 3 (j) (x 1) (x + 1) x2 + 1 9. (a) No 7. (a) (x 1) x2 + x + 1 (b) Yes (b) (x + 3) x2 3x + 9 (c) No (c) (x 5) x2 + 5x + 25 (d) Yes (d) (x + 4) x2 4x + 16 (e) No (e) (2x 3y) 4x2 + 6xy + 9y2 (f) Yes 16  5 Rational Expressions (Fractions) THE DEFINITION OF A FRACTIONAL EXPRESSION P A fraction , where Q 6= 0, such that the numerator, P, and the denominator, Q, are Q polynomials. THE FUNDAMENTAL PRINCIPLE OF FRACTIONS PK P = where Q, K 6= 0 QK Q OPERATIONS MULTIPLICATION DIVISION P R PR P R P S PS · = ÷ = · = Q S QS Q S Q R QR ADDITION AND SUBTRACTION LIKE DENOMINATORS P R P±R ± = Q Q Q UNLIKE DENOMINATORS Use the Least Common Denominator (LCD) of the fractions to rewrite them with like denominators and operate as above. COMPLEX FRACTIONS Complex Fractions are fractions whose numerators and/or denominators contain fractions. To Simplify a complex fraction write the numerator and denominator as single fractions and divide as follows: P Q P R P S = ÷ = · R Q S R R S 17  RATIONAL EXPRESSIONS (FRACTIONS) - EXAMPLES x 1 Example 1: Consider the fraction x+1 a) Evaluate it for x = 1 1 1 0 1 consider = = 0· = 0 1+1 2 2 b) Find the value(s) of x for which it is undefined consider x + 1 = 0 ) x = 1 , hence undefined for x = 1. Example 2: x2 25 (x⇠⇠ (x + 5)⇠ ⇠9 5) > > Simplify = ⇠⇠ = The Fundamental x2 4x 5 (x + 1)⇠ ⇠ (x 5) Principle of x+5 > > ; Fractions = x+1 Example 3: x y ( 1)⇠(y⇠⇠ ⇠ x) Simplify = ⇠ y x (y⇠⇠ ⇠ x) = 1 Note: All such ratios of opposites equal 1. Example 4: x (x + 1)2 x · ⇠ +⇠ (x⇠ ⇠ + 1) 1)(x Multiply and simplify · = ⇠· x · x (x + 1) x2 ⇠ +⇠ (x⇠ 1) x+1 = x Note: We may multiply by canceling common factors across the dot products. Example 5: (2 x) (x 2) (2 x) 3 Divide and simplify ÷ = · 15 3 15 (x 2) (2 x) = 5(x 2) ( 1)⇠ (x⇠⇠⇠ 2) = ⇠ 5⇠(x⇠⇠2) 1 = 5 18  RATIONAL EXPRESSIONS (FRACTIONS) - EXAMPLES Example 6: x 1 x+1 Add and simplify + = 4x + 4 4x + 4 4x + 4 +⇠ (x⇠ 1⇠ ⇠ 1) = ⇠ 4⇠ +⇠ (x⇠ 1) 1 = 4 Example 7: x 2 Subtract and simplify 2x 4 x2 2x x 2 ,LCD is = 2(x 2) x(x 2) 2x(x 2) x2 4 = 2x(x 2) 2x(x 2) x2 4 = 2x(x 2) (x⇠⇠ (x + 2)⇠ ⇠ 2) = ⇠ (x⇠⇠ 2x⇠ 2) x+2 = 2x Example 8: Simplify the complex fraction x2 9 x2 9 2 3 = 3 = x 9 ÷ x+3 x 1 x+3 3 6 + 6 2 6 (x⇠+⇠⇠ 3)(x 3) 6 ⇠ = 3 · +⇠ (x⇠ ⇠ = 2(x 3) 3) ⇠ Alternative Method: Multiply numerator and denominator by 6 (the LCD of all the fractions). ! x2 9 6 3 2(x2 9) +⇠ (x⇠ 2⇠ ⇠ 3)(x 3) != = ⇠⇠ = 2(x 3) x 1 x 1 ⇠ ⇠ (x + 3) 6 + 6· +6· 6 2 6 2 19 5.1 Rational Expressions (Fractions) – Exercises 1. Evaluate each fraction for x = 2, and find the value of x for which the fraction is undefined. 5 3x2 1 7x3 (a) (c) (e) 3x 2x 8 x2 4x 5 x+2 (b) x2 + 2 5 x+3 (d) (f) x2 1 x2 + 1 2. Simplify: 4x3 y4 24x2 54 3x2 + x 2 (a) (f) (k) 6x4 y 6x 9 3x2 5x + 2 3xy x (x 6) + 9 3x3 12x (b) (g) (l) xy + x x2 9 6x3 24x2 + 24x 5 2x 2x2 + 3x 2 (c) (h) x3 + 8 2x 5 2x 1 (m) x+2 5x 5 7x2 31x 20 (d) (i) 2a3 16 x3 x2 7x + 4 (n) 2a2 + 4a + 8 15 + 5x x2 + 2x 15 (e) (j) x2 y + y + 5x2 + 5 3x + 9 x2 7x + 12 (o) 5x + xy 3. Multiply and simplify: 27x3 24 5x 15 4x + 12 x3 x2 y 3y (a) · (d) · (g) · 9x 9x2 3x + 9 6x 18 xy 3x 3y 6y2 9xy x x2 1 (b) · (e) · x2 4a2 2a 18x2 3y3 x 1 x2 (h) · ax + 2a2 x 2a 5 x x+5 x2 x x + 2 (c) · (f) · x2 + x 20 x2 + 4x 21 5+x x 5 2x + 4 x (i) · x2 + 2x 15 x2 + 3x 28 x2 y2 y x x2 1 x2 4 3x 6 (j) · (m) · 2 · x2 2xy + y2 x + y 2x 4 x x 2 x2 + x 2 x2 1 9x2 1 x+y x3 + xy2 x2 y y3 (k) · (n) · · 3x2 + 4x + 1 3x2 4x + 1 2x2 y + 2xy2 x4 y4 y x2 + 3x + 2 x2 + 10x + 24 (l) · x2 + 5x + 4 x2 + 5x + 6 20 4. Divide and simplify: (3x)2 6x3 x2 4 x2 + 3x + 2 (a) ÷ (h) ÷ (2y) 2 16y2 x2 2x x3 + 2x2 + x 2x x2 16x2 1 4x2 7x 2 (b) ÷ (i) ÷ x + 4 (x + 4)2 4x2 + 3x 1 x2 x 2 10x + 10 2x + 2 6x3 + 7x2 6x2 + 7x (c) ÷ (j) ÷ 3x 6 3xy 6y 12x 3 36x 9 15x + 15 15 5x 2x2 + 8x 42 2x2 + 14x (d) ÷ (k) ÷ 2 x2 9 (x 3)2 x 3 x + 5x x2 + 4x 5 x 1 x3 x x x2 (e) 2 ÷ (l) ÷ x + 7x + 10 x + 4 x2 3x 4 x2 16 x2 3x 10 x2 4 3x2 2xy y2 8x 8y (f) ÷ 2 (m) 2 2 ÷ x2 5x x 2x 3x 5xy 2y 4x 8y x2 + 9x + 20 x2 25 12x4 + 15x2 4x2 + 5 (g) ÷ (n) ÷ x2 + 8x + 16 5x + 20 15x2 x 2 9x2 1 5. Multiply or divide and simplify: ✓ 2 ◆ x + 5x x2 x 20 x2 + 3x 2x2 x 4x2 + 4x + 1 4x2 2x 2 (a) · ÷ (f) · ÷ x2 25 3x + 12 3x2 27 4x2 1 x2 1 6x2 6x ✓ 2 ◆ 6x x 2 x 1 2x + 1 4x2 25 7x 1 2x + 5 (b) · 2 ÷ (g) · 2 ÷ 2 x 1 9x 4 3x + 2 3x + 3 2x 9x + 10 3x 3x 6 ✓ 2 ◆ x x 20 x2 x 2 x+1 a2 ax 4ax + 2x2 4a2 + 2ax (c) · ÷ 2 (h) · ÷ x2 25 x2 + 2x 8 x + 5x 3ax 2x2 ax x2 9a 6x ✓ 2 ◆ x x 6 x2 4x x 4 x4 8x x2 + 2x + 1 x2 + 2x + 4 (d) ÷ 2 · 2 (i) · 3 ÷ x 2 x x 2 x +x x2 4x 5 x x2 2x x 5 y2 x2 + 2x + 1 3y (e) · 2 ÷ x+1 x 1 xy y 6. Add or subtract and simplify: 5x 7x 3x + 1 5x + 2 2x + 1 (a) + (e) + 18 18 x 7 x 7 x 7 4x 24 2x2 25 + x2 (b) (f) x 6 x 6 x 5 x 5 2 4x 4 3x2 + 2x 10x 5 (c) (g) 3 2x 2x 3 x 1 x 1 3x 2x 25 3x2 x+4 (d) (h) x 5 5 x x2 1 x2 1 21  3x2 6 x 9 2x2 + x + 1 (i) + x2 + x 20 x2 + x 20 x2 + x 20 7. Add or subtract and simplify: 1 3 2x 3x 1 3 (a) (h) 3 2 x2 10x + 25 x 5 2x + 9 x 5 2 2 3 (b) (i) + 2 9x 5x x 2 x+2 x 4 2x 3 3x 2 4x 1 2 3 4 (c) + (j) 3 2 5 1+x 1 x x2 1 x 2 4x + 1 3x + 2 49x + 4 (d) (k) x 1 x2 1 x 8 x+4 x2 4x 32 x 4 + 2x (e) + x2 11 x 2 x 2 x2 4 (l) + x2 + 7x + 6 x+6 x+1 x 2 (f) 2 5x 3 + x 2x (2x 11) x2 4 4 x2 (m) + x+3 3 x x2 9 x x (g) 2 x y y x 1 3 2 (4x 1) (n) + 1 2x 1 + 2x 4x2 1 8. Simplify the complex fractions: x+4 x 1 1 1+ y (a) x + 1 (c) x (f) 7 x x+4 1 x 7 x2 1 y 1 1 y2 x x (g) x x+1 (d) y 1 1+ x+1 12 x 5x + 5y 1 1 2 (b) + 18 y x 1 x2 (e) (h) x+y 1 1 1 1 y2 x2 1 x 1+x 9 a2 2x 1 + 4x 2 x x+ (m) (i) x x 2x 2 +x 1 3 (k) x+4+ a4 x x 1 25x 2 x3 (n) 3 a 1 + 10x 1 + 25x 2 + 2 1 1 (j) a 3 (l) 1 + 1 a 1 3 + 1 1 6 2 a 1+a 1 a 22 5.2 Rational Expressions (Fractions) – Answers 5 x+1 5. (a) x 3 2y 1. (a) and 0 (e) (g) 6 x x y (b) 1 (b) 0 and -3 x 1 14 (f) (c) x (h) 11 2 (x 5)2 (c) and 4 (x + 2) (x 3) 12 (g) x (d) 11 (d) 2 and ±1 (h) 2 x2 (i) 2 x2 4 (e) 8 and 1, 5 y 5x 3 (i) 1 (e) 3 (j) (f) 1 and none (j) 1 x2 1 3x2 x 2 2y3 (f) 2 (k) (k) 1 x 1 2. (a) x+4 3x x+6 (g) 7x 1 1 3y (l) (l) (b) x+3 3 x+6 y+1 3 (h) (m) x 1 (c) 1 2 (i) 1 (m) x 3 5 1 (n) 0 (d) (n) 2 x2 2y 6. (a) x 3 5 8. (a) x 1 (e) 6 (b) 4 3 4. (a) 2 x (c) 2 (b) (f) 4x + 6 15 2 (x + 4) x+y x 3 (b) (d) 5 (c) (g) x x y x+3 (e) 0 (c) 5y (h) x+2 (d) x y 3x + 3 (f) x + 5 xy (i) x 5 (d) (e) x+3 (g) 3x 5 x y x+5 x+4 (j) (e) 3x 4 1 x 4 x+2 (h) (f) x 1 7x x+1 (f) 1 (k) x+4 1 x 1 5 (i) (g) (g) x+5 x x+2 x 5 (l) 1 2x 4 6x 7 (h) (h) x+1 7. (a) x (m) x2 2x + 4 6 (i) 1 x 3 x + 90 (i) (n) a 2 (b) x+1 (j) 3x 45x x2 + 1 2a 6 (o) (k) x + 5 41x 54 (j) x (c) a+6 30 (l) x 4 x2 3. (a) 8 x+2 (k) 2 1 (d) x a2 1 (m) x+1 (b) 2 (l) 2 x 1 3x2 (3x 1) (e) (x + 2) (c) 1 (n) x 2 (m) 2 10 5x 2 1 x 5 (d) (f) (n) 9 x 2 x+5 23  6 Roots and Radicals THE DEFINITION OF THE nth ROOT OF a p p The nth root of a is denoted by the radical, n a, where ( n a)n = a, for integer n 2. Note: n is the INDEX and a is the RADICAND of the radical. MULTIPLYING AND DIVIDING RADICALS THE PRODUCT RULE THE QUOTIENT RULE p p p s n n n a · b = ab p n a n a p = n b b ADDING AND SUBTRACTING RADICALS COMBINE LIKE RADICALS Note: Like radicals have the same index and radicand. RATIONALIZING DENOMINATORS p p x x a x a p = p ·p = a a a a p p p p x x ( a b) x( a b) p p =p p · p p = a+ b a+ b ( a b) a b p p p p Note: a b is called the CONJUGATE of a + b. THE DEFINITION OF THE RATIONAL EXPONENT p If n 2 and n a is a real number p p a1/n = n a and am/n = (a1/n )m = ( n a)m Note: all the rules for exponents hold for rational exponents. 24  ROOTS AND RADICALS - EXAMPLES p NOTE: For a > 0, the 2nd root of a, is called the square root of a, is denoted by a and is taken to be its positive (or principal) square root of a. p The negative square root of a is given by a. p p Also, a is undefined if a < 0, and 0 = 0 Example 1: Evaluate: p (a) 4 = 2, since 22 = 4 p (b) 4= 2 p (c) 4 = is undefined p (d) 3 8 = 2, since ( 2)3 = 8 Note: 3rd roots are called cube roots. p (e) 4 16 = 2, since 24 = 16 p (f) 2 ⇡ 1.414 by calculator. Note: such roots are irrational numbers. Example 2: Simplify (use the Product Rule): p p p p p p p p (a) 3 · 6 = 3 · 6 = 18 = 9 · 2 = 9 · 2 = 3 2 p p p p p (b) 3 40 = 3 8 · 5 = 3 8 · 3 5 = 2 3 5 Example 3: Simplify (use the Quotient Rule): s 14 14 p (a) = = 2 7 7 s p 49 49 7 (b) =p = 100 100 10 Example 4: Simplify : p p p (a) 4 5 + 3 5 = 7 5 p p p p p p p p p p p (b) 75 12 = 25 · 3 4 · 3 = 25 · 3 4· 3 = 5 3 2 3=3 3 Example 5: Simplify : p p p (a) 7 · 7 = ( 7)2 = 7 p p p p p p ⇠ p⇠ p ⇠ p⇠ p (b) ( 7 + 3)( 7 3) = ( 7)2 ⇠⇠ 7 · 3 + ⇠⇠ 3· 7 ( 3)2 = 7 3=4 25  ROOTS AND RADICALS - EXAMPLES Example 6: Rationalize the denominator and simplify: p p 3 3 7 3 7 (a) p = p · p = 7 7 7 7 p p p p p p p p 2 2 ( 7 3) 2( 7 3) 2( 7 3) 7 3 (b) p p = p p · p p = = = 7 + 3 ( 7 + 3) ( 7 3) 7 3 4 2 Example 7: Simplify: p (a) 93/2 = (91/2 )3 = ( 9)3 = 33 = 27 (b) ( 8)2/3 = [( 8)1/3 ]2 = [ 2]2 = 4 (c) 34/3 · 38/3 = 34/3+8/3 = 312/3 = 34 = 81 (9x)3/2 x5/2 93/2 x3/2 x5/2 (91/2 )3 x8/2 (d) = = = 33 x8/2 1/2 = 27x7/2 x1/2 x1/2 x1/2 6.1 Roots and Radicals – Exercises 1. Evaluate: p p p 4 p p3 (a) 36 (c) 25 (e) 81 (g) 225 (i) 125 p p p p p (b) 81 (d) 3 27 (f) 3 64 (h) 169 (j) 16x2 2. Evaluate with a calculator: p p ⇣p ⌘ (a) 3 (c) 4 7 (e) 6 5 3 p p p p (b) 5 (d) 3 + 11 (f) 2 7 + 4 13 3. Simplify: p p p p p p p 2 (a) 2 50 (d) 80 (g) 5 3 30 (j) 4 3 p p p p p 2 (b) 12 (e) 2 6 · 3 3 (h) 810 (k) 3 1000 p p p p p (c) 7 14 (f) 108 (i) 3 54 (l) 18x4 4. Simplify: p r p r r 48 54 5 32 5 125 (a) p (d) (g) p (j) · 3 49 4 2 2 8 r p p r 25 112 3 3 16 5 1 (b) (e) p (h) · p (k) + 9 7 2 3 54 16 4 p r r p 24 64 p 8 16a3 x (c) p (f) 3 (i) 2 · (l) p 3 27 25 2ax 26 5. Simplify: p p p p p p p (a) 2 3 + 5 3 (h) 5 3 16 2 3 54 (o) 200 12 8 + 450 p p p p p p p p (b) 9 2 4 2 (i) 27 + 2 3 75 (p) 2 20 3 45 + 180 p p p p p (c) 10 2 5 8 (j) 5 7 3 28 + 6 63 p p p p p p p p (q) 2 125 4 45 6 20 (d) 4 27 7 3 (k) 20 + 45 + 80 p p p p p p p p (r) 108 243 + 75 (e) 45 + 20 (l) 3 11 44 + 99 p p p p p p p p (f) 3 75 2 12 (m) 4 12 27 + 2 48 (s) 24 150 54 p p p p p p p p p (g) 5 15 + 48 (n) 2 63 3 28 + 4 20 (t) 8x2 x 18 + 50x2 6. Simplify: p p ⇣ p p ⌘⇣ p p ⌘ (a) 5 5 (i) 3 5 4 2 2 5+3 2 p 2 ⇣ p (b) 3 7 p ⌘⇣ p p ⌘ p p (j) 4 2+3 5 4 2 3 5 (c) 7 3 7 ⇣ p p ⌘⇣ p p ⌘ p ⇣ p p ⌘ (k) 5 2 2 11 5 2 + 2 11 (d) 2 3 2 + 2 8 ⇣ p ⌘2 p ⇣ p p ⌘ (l) 1+ 2 (e) 3 2 6 3 3 p p ⇣p p ⌘2 (f) 5 + 7 5 7 (m) 5 2 6 ⇣p ⌘ ⇣p ⌘ ⇣p (g) 5 4 5+4 p p ⌘ ⇣p p p ⌘ (n) 2+ 3 5 2+ 3+ 5 ⇣ p ⌘⇣ p ⌘ p p p p (h) 3 + 6 2 6 (o) a+x a a+x+ a 7. Rationalize the denominator and simplify: p p 1 1 3 1+ 2 (a) p (g) p (m) p 3 3 2+ 2 p 2 9 12 (b) p (h) p (n) p 5 3x 3+1 p 6 3x 3y 7 (c) p (i) p (o) p 10 5x 5y 2 2+1 p 15 3 7+ 3 (d) p (j) p (p) p 5 1+ 2 2 3 9 8 26 (e) p (k) p (q) p 2 18 5 3 5 2 3 p p p p 4 2+ 3 3 2 3 (f) p (l) p p (r) p p 3 12 3 2 2 2+ 3 27  p ⇣ p ⌘ ⇣p ⌘ p 8 6 3+ 5 5 2 2x xy (s) p (w) p 8+6 (u) p 2 xy y 5 5 p p p 25 3 4 2 x 1 (t) p p (v) p p (x) p 7 3 5 2 x+ y x+1 1 8. Simplify: ✓ ◆ 1/2 p (a) 43/2 9 (l) 35/2 48 (g) 25 (b) 272/3 (m) 8x3 y6 1/3 (h) 21/2 · 23/2 (c) 253/2 5/3 4/3 (4x)3/2 y5/2 (i) ( 3) · ( 3) (n) (d) ( 8)5/3 (xy)1/2 51/2 ✓ ◆ (e) 16 1/4 (j) 27x 3 1/3 53/2