Unit 2 Rational Functionals Exercises MHF 4UI Page Exercises 2.: Division of Polynomials. Divide, assuming the divisor is not equal to zero. a) x 3 + 2x 2 7x + 4 ) x + ) b) 3x 4 4x 2 2x + 3 ) x 4) 7. *) If fx) = dx)qx) + rx) and degdx)) = 3 and degfx)) = 7, what are the possible degrees of rx)? State examples of the equation for each degree. 8. **) Note that the sum of the powers of 2 is less than the next power of 2. That is c) d) e) x 3 ) x ) x 4 ) x ) x 5 ) x ) + 2 = 3 = 4 + 2 + 4 = 7 = 8 + 2 + 4 + 8 = 5 = 6 f) x 3 4x 2 2x + ) x 3) 2. For fx) = x 2)x 2 + 3x + 4) + 6 find: Explain why this is true. Hint: Divide x n by x and substitute a special value for x. a) The divisor. b) The quotient. c) The remainder. d) The dividend. 3. In each of the following, obtain a quotient and remainder, leaving your answers in the form fx) = dx)qx) + rx). a) b) c) d) e) f) 5x 3 2x 2 9x + 5 ) x + ) x 4 + 3x 3 2x 2 + 7x + 2 ) 2x 3) 2x 4 + 5x 3 4x 2 3x + 4 ) x 2 + 3x + 5 ) x 3 + kx 2 + x + ) x ) x 5 + 6x 2 x + ) x 2 7x + ) x 3 + x 2 + x + ) x + k) 4. Divide x 3 + a + b)x 2 + ab + c)x + ac by x + a. 5. *) If fx) = dx)qx) + rx), what conditions on fx) isare) required for rx) to vanish? 6. *) If fx) = dx)qx) + rx) and degdx)) = 2, what are the possible degrees of rx)?
Unit 2 Rational Functionals Exercises MHF 4UI Page 2 Exercises 2.2: The Remainder Theorem. Explain how to find the remainder of x 3 +4x 2 2x+3 divided by x 4. 2. Find the remainder of each of the following divisions. a) x 3 + 4x 2 + 2x 7 ) x + ) b) c) d) x 4 3x 2 3x + 5 ) x 2) x 7 2x 5 + 3x 3 4x + 5 ) x + 5) x 4 + 6x 3 + 9x 2 7x + 3 ) x 9) 7. Let fx) be a polynomial that has remainder 3 when divided by x 2. Find the remainder when each polynomial is divided by x 2. a) fx) + 3 b) 3x + )fx) c) fx) + 5x + 2 d) [fx)] 2 + 2fx) 8. Find the remainder when a polynomial fx) is divided by: e) f) 3x 5 + 2x 2 + 3 ) x 5) x 4 2x 3 + 3x 2 x + 5 ) x + ) a) ax + b, a 0 b) x a)x b) g) x 3 + kx 2 x + 4 ) x + ) h) x 3 + ax 2 + bx + c ) x ) 3. Find the value of k in each of the following. 9. Outline an approach for finding the remainder of an n-th degree polynomial fx) divided by x a )x a 2 ) x a k ) where k < n. a) When x 3 + kx 2 3x + is divided by x, the remainder is 2. b) When x 3 kx 2 x + 4 is divided by x 2, the remainder is 9. c) When x 4 + kx 3 + 4x 2 + 2 is divided by x + 3, the remainder is 4. 4. When x 3 + ax 2 + bx + 3 is divided by x 5, the remainder is 2 and when it is divided by x + 3, the remainder is 7. What are the values of a and b? 5. When ax 4 + bx 3 + 3x 2 2x + is divided by x +, the remainder is 3 and when it is divided by x 2, the remainder is 5. What are the values of a and b? 6. Find the remainder for each of the following. a) b) c) x 5 + 4x 3 2x 2 + ) x + )x + 2) x 5 2x 4 + 3x 3 2x 2 + x 5 ) x )x 3) x 6 + 2x 2 ) x 2)x + 3)
Unit 2 Rational Functionals Exercises MHF 4UI Page 3 Exercises 2.3: The Factor Theorem. Given the polynomial fx) = x 3 + 5x 2 + 6x + 8: a) Show that f 4) = 0. b) Using long division, show that x + 4 divides fx). 2. If x 2 is a factor of fx), what is the value of f2)? 3. If f 3) = 0, then what must be a factor of fx)? 4. Which of the following polynomials have x + as a factor? a) x 3 2x 2 + 3 b) x 3 x 2 + x + 3 c) x 4 + 6x 3 x 2 5x + d) x 3 + 5x 2 3x 7 e) 4x 3 + 4x 2 2x + 3 f) 2x 5 x 4 x 3 + x 2 + 0x + 5. Given the polynomial fx) = 4x 3 + 7x 2 + 8x + 9: a) Show that f 3) = 0. b) Find a linear factor of fx). b) Use long division to find the remaining factors of fx). 6. Find k so that x + 2 is a factor of x 3 + 2kx + 3x 4. 7. Find k so that x 4 is a factor of 2x 3 kx x +. 8. Factor completely by finding one factor, then using long division to find the remaining factors. a) x 3 4x 2 + x + 6 e) x 4 x 3 3x 2 + x + 2 f) x 3 + 3x 2 + 6x + 4 g) 2x 3 x 2 7x + 2 h) x 4 5x 2 + 4 9. Using a sum or difference of cubes, factor the following. a) x 3 8 b) x 3 + 343 c) 8a 3 64b 3 d) 625z 3 40 e) a + b) 3 b 3 f) a + b + ) 3 a + b ) 3 0. Use the Factor Theorem to show that 2x 3 x 2 + 5x + 8 is not divisible by x 2 + x 6.. Show that x y is a factor of each of the following. For each exercise, find the other factor. a) x 4 y 4 b) x 5 y 5 c) x 6 y 6 d) x n y n 2. *) When is x + y a factor of x n + y n? 3. Completely factor x 5 + y 5. b) x 3 3x 2 4x + 2 c) x 3 x 2 + 35x 25 d) 2x 3 5x 2 4x + 3
Unit 2 Rational Functionals Exercises MHF 4UI Page 4 Exercises 2.4: Solving Polynomial Equations. Write a polynomial equation with zeros at x =, 0, -. 2. Write a polynomial equation with zeros at x = 3, 5, -2. 3. Find all polynomial equations with zeros 0,, 2. 4. Find all polynomial equations with zeros, 2. 5. Solve the following equations: a) x + )x 2) = 0 b) x )x 3) = 0 c) 2x + )x 5) = 0 d) x 3) 2 = 0 e) x 2 + 2x + = 0 f) x 2 + 7x + 2 = 0 g) x 2 + 3x 30 = 0 h) 2x 2 5x + 3 = 0 i) 6x 2 + 7x + 2 = 0 j) 0x 2 + 29x + 0 = 0 6. Solve the following equations. a) x 2 + 5x + 2 = 0 b) 2x 2 7x + 2 = 0 c) 3x 2 6x 4 = 0 d) 7x 2 4x 3 = 0 e) 4x 2 8x 3 = 0 f) x 2 25x + 6 = 0 7. Solve the following equations. a) x + 3) 2 = x + ) 2 b) a 2 + a 2 + 3a = 5a 2 c) m 5) 2 = m 2 d) 5y = y 2 + 3y e) z 3 + 3z = 5z 2 + 8z 3 f) p 3 + 4p = p + ) 3 8. Solve the following equations. a) b) c) d) x + + x = 2 2x + 4x 3 = 5 x x + + 3 x + 2 = 6 2x x 4x 2x + = 3 9. Solve the following equations. State any necessary restrictions. a) x 2 + mx + n = 0 b) ax 2 + bx = 0 c) ax 2 + b = 0 d) kx 2 + k + )x + k + 2 = 0 0. Write and solve an equation that finds three consecutive integers that have a product of 76.. Find the value of k such that x 3 +5x 2 +kx+ gives the same remainder when divided by x and x 2. 2. The tensile strength of a new composite material is modeled by the function S = 8 + 5T T 3, where T is the annealing temperature in 000 K.
Unit 2 Rational Functionals Exercises MHF 4UI Page 5 a) Graph the function. b) What restricted domain of the function makes physical sense? c) What is the approximate maximum strength of the material? d) What is the minimum strength of the material? 3. **) Show that the product of two consecutive natural numbers can never be a perfect square. Exercises 2.5: Solving Polynomial Equations, Part 2. Solve the following equations: a) x + )x + 2)x + 3)x + 4) = 0 b) x )x 2)x 3)x 4) = 0 c) x + )x )x + 2)x 2) = 0 d) x + )x 2 + 4x + 2) = 0 e) x 3 + 3x 2 x 3 = 0 f) 2x 3 3x 2 2x + 3 = 0 g) 4x 4 2x 3 3x 2 + 98x 24 = 0 h) x 4 + 5x 3 + 5x 2 5x 6 = 0 i) x 4 + 6x 3 + 3x 2 26x 24 = 0 j) 30x 3 53x 2 2x + 6 = 0 2. Solve the following equations: a) x 3 + 9x 2 + 9x + 3 = 0 b) x 3 + 0x 2 50x 24 = 0 c) x 4 3x 3 2x 2 72x 28 = 0 3. Explain why every cubic equation has at least one real root. 4. Find the complete family of polynomials that have zeros and 2. Then find two different cubic functions that pass through the point 3, 5).
Unit 2 Rational Functionals Exercises MHF 4UI Page 6 Exercises 2.6: Graphs of Rational Functions. Graph the following functions. For each graph, state the domain and range, the y-intercept and any asymptotes. For parts f) amd i), consider the three cases n > 0, n = 0 and n = k < 0. Exercises 2.7: Graphs of Rational Functions, Part 2. Graph the following functions. For each graph, state the domain and range, the y-intercept and any asymptotes. a) y = x a) y = x 2 b) y = x 2 c) y = x 5 d) y = x + 2 e) y = x + 3 f) y = x + n g) h) y = 5 x y = 3 x i) y = n x + n 2. Graph the function y = / x. What are the vertical and horizontal asymptotes, the domain and range of the function? Repeat the exercise if the function undergoes the transformation x x a and y y b. 3. Plot the function y = /x n for n =, 2, 3, 4, 5, 6 on the same graph. Find domain, range, vertical and horizontal asymptotes. Summarize your findings and hypothesize for n N. 4. Draw the graph that has horizontal asymptote at y = 5 and vertical asymptote at x = 3, with y- intercept x = 5. If the graph is a translation of the function y = a/x, find the equation of the graph and the x-intercept. b) y = c) y = d) y = e) y = x 2) 2 x 2)x + ) x + 2) 2 5 x 2 3x + 2 f) y = x + 3) 2 g) y = h) y = x 2 + 7x + 2 3 x 2 4x 2 i) y = x n) 2 2. Relate the functions fx) = /x 2 and gx) = x a) 2 + b by a simple transformation. How are the graphs related? 3. Graph the following functions using a table of values. State the domain, range, any intercepts and any asymptotes. a) y = x 2 + x + b) y = x 2 + c) y = x 2 + 2x + 5
Unit 2 Rational Functionals Exercises MHF 4UI Page 7 4. Describe any qualitative differences between the graphs in Exercises #, #2, and #3. Why do these differences exist? 5. **) The Witch of Agnesi is constructed by considering a circle of radius a and centre 0, a) and drawing a line OA from the origin to the line y = 2a which is point A. The x-coordinate of the Witch is given by the x-coordinate of the point A and the y- coordinate of the intersection of OA with the circle. a) Draw a diagram that represents the description given. b) Show that the coordinates of the Witch are given by x = 2a cot θ and y = a cos 2θ), where θ [0, π]. c) Eliminate θ in the two equations to find the equation of the Witch y = 8a 3 /x 2 + 4a 2 ). d) Draw a graph of the Witch. e) State the domain, range, any intercepts and asymptotes. Exercises 2.8: Graphs of Rational Functions, Part 3. Graph the following functions. For each graph, state the domain and range, any intercepts, holes and any asymptotes. a) y = b) y = c) y = x x + )x ) 3x x + )x 2) 2x + x 3)x ) d) y = x xx + ) e) y = 2x xx + 2) f) y = 3 x + x 2 2. Graph the following functions, simplifying first if necessary. For each graph, state the domain and range, any intercepts, holes and any asymptotes. a) y = b) y = c) y = d) y = e) y = f) y = x + x )x + 3) x + 2 x 2)x + )x + 4) 2x x )2x 3)x 2) x ) 2 xx )x 2) 2 x + ) 3x + x + ) 2 x ) 3 x 2 x ) 3 x + ) 2 x + 2) 2 2x ) 2 x 2) 3
Unit 2 Rational Functionals Exercises MHF 4UI Page 8 3. Graph the following functions, simplifying first if necessary. For each graph, state the domain and range, any intercepts, holes and any asymptotes. a) y = b) y = c) y = d) y = e) y = f) y = x + x 2 + 3x + 2 x 2 x 3 + 2x 2 5x 6 x 2 2x 3 x 2 8x + 4 x 2 + 6x 3 + 3x 2 + 25x 2 2x 5 x 3 + 7x 2 36 2x 2 + 3x 2 x 4 3x 3 9x 2 40x 84 Exercises 2.9: Graphs of Rational Functions, Part 4. Graph the following functions. For each graph, state the domain and range, any intercepts, holes and any asymptotes. a) y = + x + b) y = 3 + c) y = 2 + 2 x + )x 2) x + x + 3)x 2) d) y = + x xx 2) e) y = + 2x x )x + 2) f) y = 4 + x x )x + ) 2. Simplify each function using factoring and/or long division and then graph each function to determine its key features. a) y = x + x + 2 b) y = x x 2 c) y = d) y = e) y = f) y = x + )x 2) x )x + 2) x 2)x 3) x 2)x + ) x + 2)x + ) 2 x )x + )x 2) x )x + )2 xx + )x + 2)
Unit 2 Rational Functionals Exercises MHF 4UI Page 9 3. Simplify each function using factoring and/or long division and then graph each function to determine its key features. a) y = x2 2x + x 2 + 7x + 2 b) y = x2 + 5x 6 x 2 7x 44 c) y = x2 3x 0 2x 2 + 5x 3 d) y = 2x3 5x 2 x + 6 x 3 3x 2 + 3x e) y = f) y = x 3 x 3 + 4x 2 4x 6 x 2 6x 2 + x 0 Graphs of Rational Func- Exercises 2.0: tions, Part 5. Graph the following functions. For each graph, state the domain and range, any intercepts and any asymptotes. a) y = 2x + + x + b) y = x + 3 + 2 x 2 c) y = x 4 + d) y = 3x 5 + 3x x )x + 2) x x + )x ) e) y = x 6 x + x + 2) 2 f) y = x 2 x + ) 2 x ) 2 x 2) 2. Simplify each function using factoring and/or long division and then graph each function to determine its key features. a) y = x2 x + b) y = c) y = x + 2)x + ) x 2 x )x + ) x 3 d) y = x2 x ) x + )x ) e) y = x )x + )2 x 2)x ) f) y = x + ) 2 x 2) 3 x 2)x + 3)x + 4) 2
Unit 2 Rational Functionals Exercises MHF 4UI Page 0 3. Simplify each function using factoring and/or long division and then graph each function to determine its key features. a) y = x2 2x 3 x 2 b) y = x2 5x 24 x c) y = 0x2 7x 20 3x 2 d) y = 2x3 + x 2 25x + 2 x 2 6x + 8 e) y = x 3 2x 2 + 23x 24 f) y = 30x4 + 2x 3 + 4x 2 64x 96 x 3 x 2 4x + 24 Exercises 2.: Review. Divide the following. a) 2x 3 + 4x 2 7x + ) x + 4) b) 6x 4 2x 3 + 4x 2 3x 7 ) x + 7) c) 3x 3 + 4x 2 5 ) x 2) d) 6x 4 + 2x + ) 3x + ) e) 7x 4 + 5x 3 + 3x 2 9x + 4 ) x 2 + 3x 4) 2. Divide the following. a) 6x 3 x 4 + 2x + 3 ) x + ) b) x 3 + x 2 x + 5 ) x + ) 4. *) Graph the function y = x 3 /x ) by a table of values and by using long division. To what function do the asymptotes appear to be converging for large positive and negative values of x? c) d) e) x 4 + 3x 3 7x 2 + 2x + 3 ) x + 5) x 6 + x 4 x 2 + 3 ) 2x ) x 6 + 3x 2 x + ) x 2 + 5x 3) 3. Find the remainder of each of the following divisions. a) b) c) d) x 4 6x 3 + x 2 + 5 ) x + 4) x 7 + 5x 2 + 3 ) x 3) 3x 5 + 2x 2 + 5x + ) x )x + 2) 3x 4 + 2x 3 + 5x 2 + 2x + ) x 2 + 3x + 2) 4. Find the value of k so that when x 3 + 7x 2 kx + 3 is divided by x 4, the remainder is 3. 5. If x + is a factor of fx), then evaluate f ). 6. Show that x + is a factor of x 3 x 2 7x 5.
Unit 2 Rational Functionals Exercises MHF 4UI Page 7. Factor the following. a) x 3 + 6x 2 + x + 6 b) x 3 + 4x 2 + x 6 c) x 4 + 3x 3 + x 2 3x 2 d) 2x 4 x 3 7x 2 + 86x + 20 8. Factor 27x 3 8y 6 using difference of cubes. 9. Solve the following equations. a) x 3 3x 2 = 0 b) x 3 2x 2 x + 2 = 0 c) x 3 5x 2 8x + 2 = 0 d) 6x 3 + 5x 2 7x 6 = 0 e) x 3 + 3x 2 40x = 0 f) x 3 28x = 0 g) h) x x + + 2x + x + 2 = 5 x + 3 2 x + = 4 x + 2 0. Graph the following. For each function, state the domain, range, any intercepts, holes and asymptotes. a) y = x b) y = x c) y = 3 x + 2 d) y = 5 x + 3 e) y = f) y = 4 x + )x 5) 3x + 2 2x )x + 3) g) y = 6x2 7x 3 x )x 2) 2 h) y = x n x ) 2n, n Z. Graph the following. For each function, state the domain, range, any intercepts, holes and asymptotes. a) y = 2x x + 4 b) y = x 8 x 2 c) y = d) y = e) y = x + x x + 5) 2 x + 2)x + 5) x 2 x 2 + 7x 8 f) y = 2 x + 3 + 4 g) y = 2x x 2 + h) y = x3 + 2x + 6 x + )x 2) i) y = x n x ) n, n Z 2. Draw the graph similar to y = a/x b) that has horizontal asymptote y = 2, vertical asymptote x = and passes through the point 4, 7).
Unit 2 Rational Functionals Exercises MHF 4UI Page 2 Exercises 2.: Solutions Division of Polynomials. Division. a) x + )x 2 + x 8) + 2 b) x 4)3x 2 2x 0) 37 c) x )x 2 + x + ) d) x )x 3 + x 2 + x + ) e) x )X 4 + x 3 + x 2 + x + ) f) x 3)x 2 x 5) 4 2. Classifying division. a) x 2 b) x 2 + 3x + 4 c) 6 d) x 3 + x 2 2x 2 3. Division. a) x + )5x 2 7x 2) + 7 b) 2x 3) 2 x3 + 9 4 x2 + 9 8 x + 3 ) 6 + 37 6 c) x 2 + 3x + 5)2x 2 x ) + 35x + 59 d) x ) x 2 + k + )x + k + 2) ) + k + 3 e) x 2 7x+)x 3 +7x 2 +48x+335)+2298x 334 f) x + k) x 2 + k)x + k 2 k + ) + k + k 2 k 3 4. x + a)x 2 + bx + c). 5. For rx) to vanish, dx) must be a factor of fx). 6. The possible degrees of rx) are zero and one. 7. The possible degrees of rx) are zero, one and two. 8. Notice that x n )/x ) = x n + x n 2 + + x 2 + x +. When x = 2, this means that 2 n = 2 n + 2 n 2 + + 2 2 + 2 +. Therefore, we see that when n =, the left side is 2, and the right side is. When n = 2, the left side is 4 = 3, and the right side is 3 = 2 +. And so on. Exercises 2.2: Solutions The Remainder Theorem. r = f4) = 23 2. Finding remainders. a) 6 b) 3 c) 72225 d) 604 e) 9322 f) 2 g) 4 k h) + a + b + c 3. Finding k. a) k = 3 b) k = 4 c) k = 5 27 4. a = 263 30, b = 559 30 5. a = 7 6, b = 6 6. Finding remainders. a) rx) = 65x + 59 b) rx) = 55x 59 c) rx) = 35x + 34 7. Finding remainders. a) 6 b) 2 c) 20 d) 5 8. Finding remainders. a) r = f a ) b b) rx) = fa) fb) a b x afb)+bfa) a b
Unit 2 Rational Functionals Exercises MHF 4UI Page 3 Solutions The Factor Theo- Exercises 2.3: rem 0. x 2 + x 6 = x + 3)x 2); f 3) 0 and f2) 0 therefore not divisible.. f 4) = 4) 3 + 5 4) 2 + 6 4) + 8 = 0 2. f2) = 0 3. x + 3 is a factor of fx) 4. Factors. a) Factor. b) Factor. c) Factor. d) Factor. e) Not factor. f) Not factor.. Other factors. a) x 3 + yx 2 + y 2 x + y 3 b) x 4 + yx 3 + y 2 x 2 + y 3 x + y 4 c) x 5 + x 4 y + x 3 y 2 + x 2 y 3 + xy 4 + y 5 d) x n + x n 2 y + x n 3 y 2 + + xy n 2 + y n 2. When n is an odd positive integer. 3. x 5 + y 5 = x + y)x 4 x 3 y + x 2 y 2 xy 3 + y 4 ) 5. fx) = x + 3)4x 2 + 5x + 3), quadratic factor has complex roots. 6. k = 9 2 7. k = 25 4 8. Factoring. a) fx) = x + )x 2)x 3) b) fx) = x + 2)x 2)x 3) c) fx) = x )x 5) 2 d) fx) = x + )2x )x 3) e) fx) = x )x + ) 2 x 2) f) fx) = x + )x 2 + 2x + 4) g) fx) = x 2) 2x + 3 + 7 ) 2x + 3 7 ) h) fx) = x + 2)x 2)x + )x ) 9. Factoring a sum or difference of cubes. a) x 2)x 2 + 2x + 4) b) x + 7)x 2 7x + 49) c) 8 a 4 /3 b ) a 2 + 4 /3 b + 4 2/3 b 2) d) 55z 2)25z 2 0z + 4) e) aa 2 + 3ab + 3b 2 ) f) 23a 2 + 3b 2 + 2ab + )
Unit 2 Rational Functionals Exercises MHF 4UI Page 4 Exercises 2.4: Solving Polynomial Equations. x )x)x + ) = 0 2. y = x 3)x 5)x + 2) = 0 3. kx a x ) b x 2) c = 0 4. kx ) a x 2) 2 = 0 5. Solving equations. a) x =, x = 2 b) x =, x = 3 c) x = 2, x = 5 d) x = 3 e) x = f) x = 3, x = 4 g) x = 5, x = 2 h) x = 3 2, x = 2 i) x = 2 3, 2 j) x = 2 5, 5 2 6. Solving equations. a) x = ) 2 5 ± 2 b) x = ) 4 7 ± 33 c) x = ) 3 8 ± 76 d) x = ) 7 7 ± 70 e) x = ) 28 8 ± 6279 f) x = 2, x = 3 7. Solving equations. b) x = ) 20 3 ± 9 c) No solution. d) x = ) 4 3 ± 7 9. Solving equations. a) x = 2 m ± ) m 2 4n, m 2 4n b) x = 0, x = b a, a 0 c) x = ±, ab < 0, a 0 d) x = 2k b a k ± 3k 2 6k + 3k 2 6k + 0 ), k 0, 0. Solutions of x 3 + 3x 2 + 2x 76 = 0 are x =, x = 2 and x = 3.. k = 22 2. 0 T 2.803, the maximum strength is S 2.303 and the minimum is S = 0. 3. Let n, n + be two consecutive natural numbers. Suppose that nn+) = k 2, where k Z. Then n 2 + n k 2 = 0. The discriminant is D = + 4k 2. Since D can t be an integer, then n can t be an integer, which means that there can be no two consecutive natural numbers that are perfect squares. a) x = 2 b) No solution. c) m = ) 2 ± 3 d) y = 0, y = 2 e) z = 0, z = ) 4 5 ± 09 f) p = 0, p = 3 8. Solving equations. a) x = ) 2 ± 5
Unit 2 Rational Functionals Exercises MHF 4UI Page 5 Exercises 2.5: Solving Polynomial Equations, Part 2. Solving equations. a) x =, x = 2, x = 3, x = 4 b) x =, x = 2, x = 3, x = 4 c) x = ±, x = ±2 d) x =, x = 2 ± 2 e) x = 3, x =, x = f) x =, x =, x = 3 2 g) x = 2, x = 3, x = 4, x = 4 h) x = 3, x = 2, x = 2, x = i) x = 4, x = 3, x =, x = 2 j) x = 2, x = 2 3, x = 8 5 2. Solving equations. a) x = 3, x = 3 ± 8 b) x = 4, 7 ± 43 c) No solution. 3. If the polynomial has real coefficients and the roots are complex, the roots must be in conjugate pairs. Since a cubic has three roots, the third root must be 4. y = kx ) a x 2) 3 a, where a {, 2}. y = 5 2 x )x 2)2, y = 5 4 x )2 x 2). Exercises 2.: Solutions Review. Division. a) x + 4) x 2 4x + 9 ) 35 b) x + 7) 6x 3 44x 2 + 32x 287 ) +5302 c) x 2) 3x 2 2x 4 ) 3 d) 3x + ) 2x 3 2 3 x2 + 2 9 x + 6 ) + 27 27 e) 2. Division. x 2 + 3x 4 ) 7x 2 6x + 79 ) 294x + 320 a) x + ) x 3 + 7x 2 7x + 9 ) 6 b) x + ) x 2 ) + 6 c) x + 5) x 3 2x 2 + 3x 3 ) + 68 d) 2x ) 2 x5 + 4 x4 3 8 x3 + 5 6 x2 + 32 x + ) + 203 64 64 e) x 2 + 5x 3 ) x 4 5x 3 + 28x 2 55x + 862) + 4776x + 2587 3. Remainders. a) 49 b) 2235 c) 36x 25 d) 46x + 45
Unit 2 Rational Functionals Exercises MHF 4UI Page 6 4. k = 44 5. f ) = 0. 6. Evaluate f ) = 0. 7. Factoring. a) x + )x + 2)x + 3) b) x )x + 2)x + 3) c) x )x + ) 2 x + 2) d) x + 2)2x + 3)x 4)x 5) 8. 3x 2y 2 )9x 2 + 6xy 2 + 4y 4 ) 9. Solving. a) x = 4 x = x = 3 b) x = x = x = 2 c) x = x = 2 x = 6 d) x = 2 x = 3 x = 3 2 e) x = 0 x = 5 x = 8 f) x = 0 x = ±8 2 g) x = 5 ± 73 4 h) x = 23 ± 89 0