UL - Review of Distributive Law and Factoring. Expand and simplify. a) (6mn )(-5m 4 n 6 ) b) -6x 4 y 5 z 7 (-x 7 y 4 z) c) (x 4) - (x 5) d) (y 9y + 5) 5(y 4) e) 5(x 4y) (x 5y) + 7 f) 4(a b c) 6(4a + b 6c) g) (x 9) (4x + ) + h) [5 + 4(x 7)] i) -[(x 5) 4(x )] j) [x + (x )] x(x + 4) k) (x 7)(x + ) l) (m + 5) m) 5(y 5)(y + 5) n) (t )(t + 4) (t + 6)(t + ) o) 5x(x 4x + 7) p) (y + 7x)(y x) q) 4(x xy) + (x + y). Expand and simplify (x + )(x + x + 4) fully.. Factor, if possible. a. x 5x 4 b. x 8x 5 c. t 9t d. r r 4 e. n n 0 f. r 7r 0 g. w 0w 6 h. x 9x 4 i. m 0m 4 4. Factor, if possible. a. y y 0 b. x 7x 8 c. t t 8 d. n 0n 4 e. r 7r 0 f. x 8x 0 g. x 0 h. w w 0 i. r r 0 j. y 0y 6 k. n 4n 5 l. 8 7y y m. 6 6x x 5. Factor, if possible. a. x xy 5y b. a 4ab 77b c. c cd d d. x 5xy 6y e. x 4xy 6y f. p 4pq q 6. Factor if possible. a. x x 9 b. 5y 40y 60 c. 4t 8t 60 d. 6x 8x 4 e. ax 0ax 4a f. x 8x 7x g. x x 56 h. 5w 0w 60 i. x x x 7. Factor, if possible. a. m 8m 80 b. m m c. x x 5 d. r 7r 4 e. y 7y 7 f. x 6x 6 g. y y 4 h. m 7m 6 j. x 0 k. w w 0 l. r r 0 m. y 0y 6 n. n 4n 5 o. 8 7y y p. 6 6x x
Answers UL:. a) -0m 5 n 9 b) 8x y 9 z 8 c) x + 7 d) 5y y + 5 e) x 5y + 7 f) -8a 0b + c g) x 9 h) x 69 i) 8x 0 j) x + x - k) x - 6x 7 l) m + 0m + 75 m) 0y 5 n) t t 6 o) 5x 0x + 5x p) y + xy 4x q) 5x 0xy + y. x + 5x + 0x +. a) x 4 x b) x 5 x c) n.p. d) r 6 r 7 e) n 6 n 5 f) r r 5 g) w w 8 h) n.p. i) m 4 m 6 4. a) y 4 y 5 b) x 9 x c) n.p. d) n n e) n.p. f) x x 0 5. a) x 7y x 5 y b) a b a 7 b c) c d c d x 4y x 9 y e) n.p. d) f) p q p 6 q x x b) 5 y 6 y 6. a) c) 4 t 5 t d) 6 x 4 x e) a x x f) x x x 6 g) x 7 x 4 h) 5 w 6 w i) x x x 7. a) m 0 m 8 b) m 4 m c) n.p. d) r r 4 e) y 9 y 8 f) x x 8 g) n.p. h) n.p. i) x x 7 j) w w 0 k) r 5 r 6 l) y y 8 m) n.p. n) y 8 y o) 8 x x p) (8+x)(-x)
UL - Continued Special Patterns. Factor, if possible. a) x 9 b) y 6 c) z 8 d) 5a 6 e) 64t f) 6 49a g) 49 x h) 5x 64y i) k) 4t 9s j) 00 p q 6 8y l) 5b a. State whether each trinomial is a perfect square trinomial. If it is, factor it. a) x 6x 9 b) y 0y 5 c) x 8x 4 d) 4t 4t e) 6t 4t 9 f) x 49 4x x g) 6t 64t h) 9x 4x 6 i) 4 8r 49r j) 8x 7xy 64y k) m m l) 9a ab 4b. Factor fully, if possible. a) y 44 b) 5x 5y c) e) 9a 4a 6 d) x y 6 f) x 6x Answers:. a) ( x )( x ) b) ( y 4)( y 4) c) not possible d) (5a 6)(5a 6) e) ( 8 t)( 8 t) f) (6 7 a)(6 7 a) g) not possible h) (5x 8 y)(5x 8 y) i) (t s)(t s) j) (0 p q)(0 p q) k) (6 9 y)(6 9 y) l) (5 b a)(5 b a). a) yes, (t ) e) yes, ( 8 ) yes, t h) yes, ( x ) b) yes, (4t ) (x 4) (m ) l) yes, ( y 5) c) no d) yes, f) yes, (7 x) g) yes, i) yes, ( 7 r) j) no k) (a b). a) ( y )( y ) b) not possible c) ( x 4)( x 4) e) not possible f) ( x ) (a 4) d)
UL - Solving Quadratics. Solve. a) ( x )( x ) 0 b) ( x )( x ) 0 c) ( x 5)( x 5) 0 d) ( x )( x ) 0 e) (x )( x ) 0 f) (x 4)(x ) 0 g) xx ( 9) 0 h) x(4 x) 0. Write each equation in the form ax bx c 0. a) x 6 x b) y y c) e) ( z ) 4 4m z d) m f) ( x ) 4 ( x ) x. Solve by factoring and check. a) n 7n 0 b) y y 0 c) x x 6 0 d) a 8a 6 0 4. e) 0 p p 5 f) m 7m 8 Solve by factoring a) c) e) a a 0 b) s 4s 0 t t 5 0 d) x 7x 6 0 0 4m 4m f) 0y 6y 6 5. Solve by factoring. a) x x 0 b) y c) e) m m 0 y 0 d) 5n 8n 0 5t 0t 0 f) 0 4x x 6. Solve by factoring. a) 0 = x + 4x x b) 0 = x 4 65 c) 0 = x + x - 8x 7 d) 0 = x 4 7x 5x e) x 6x + 9x = 0 f) x + 5x x = 5 g) x 4 = 9x Answers:. a) x = - or x = - b) x = - or x = c) x = 5 d) x = or x = - e) x or x = f). a) 4 x or x g) x = 0 or x = -9 h) x = 0 or x = x x 6 0 b) y y 0 c) z 4z 0 d) x x 0 e) 4m m 0 f) x x 0. a) n = - or n = -4 b) y = or y = c) x = - or x = d) a = 4 e) p = -7 or p = 5 f) m = - or m = 9 4. a) a = - or -5 or t d) a b) s = or x or x = - e) s c) t = m or m f) y = or y 5 5. a) x = 0 or x = - b) y = 0 or y = c) m = 0 or m d) n = 0 or 8 n 5 e) t = 0 or t = 4 4 f) x = 0 or x 6. a) x = 0, x =, x = -7 b) x = 5, x = -5 c) x =, x = -, x d) x = 0, x = 5, x e) x = 0, x = f) x =, x = -, x = -5 g) x = 0, x =, x = - More Questions on Next Page 4
UL Continued. Solve using the quadratic formula. Give exact answers. a) 6x 7x = 0 b) x + 6x + = 0 c) x + 6x + = 0 d) x + 7x + = 0 e) X + 6x + 4 = 0. Determine the exact values of the x-intercepts of each quadratic function. Then approximate the roots to the nearest hundredth. a) Y = x + 5x + b) F(x) = x 6x + 7 c) G(x) = x +x + 6 d) H(x) = 4 x -5x + 5. Use the discriminant to determine the number of roots for each quadratic equation. a) x x + = 0 b) x 6x + = 0 c) x 5x + 7 = 0 d) x + 5.5x +.5 = 0 e) 5x 0x + 5 = 0 4. Determine the value(s) of k for which the quadratic equation x + kx + 4 = 0 will have each number of roots. a) One distinct real root (two real, equal roots) b) Two distinct read roots c) No real roots 5. The height of a football can be modelled by the function h(t) = -4.9t +.8t +.5, where t is the time, in seconds, since the ball was thrown, and h is the height of the ball, in metres, above the ground. Determine how long the football will be in the air, to the nearest tenth of a second. AnswersUL:. a) x =, x = =. a) x = 5± 7 4 b) = x = ± 6 c) x = ± d) x = 7± 6 b) x = ± c) x = ± d) x = 0± 0 e) x = ± 5. a) distinct real roots b) distinct real root c) no real roots d) two distinct real roots e) distinct real root 4. a) k = 4, k = -4 b) k > 4 or k < -4 c) -4 < k < 4 5. about 4.5 seconds 5
UL - Solving Factorable Polynomials. Solve. a) x 4x + x b) x -6x -x + 9 c) 4x + 8x x d) x 6x +0x 0 e) x 5 -x -x +4 f) x 4-4x -8x +6x. Solve a) x x + b) x -9x c) 6x +x-7 d) 0x -x +8x e) x -x +4x- f)4x +0x+5 g) 5x -0x+45 h) 4x +9x-5 i) 4x 4 +8x +49 Answers UL:. a) x = 4 b) x = {, } c) x = {, ± } d) x= e) x = {±, } f) x = {0, ±}.a){,} b) {0,} c) { 7,} d){8, } e) {} 5 f) { 5 } g) {} h) {, 5} i) no solution 4.a) 00 b) 00 c) 0. Use difference of squares factoring to evaluate each of the following. a) 5-49 b) 7 - c) - UL4 -Multiplicity and Sketching Polynomials. For the following graphs: i) State the x-intercepts ii) Determine the number of times each factor occurs in the factored form of the function. 6
. Without using technology, match each graph with the corresponding function. Justify your choice.. For each graph of a polynomial function shown to the right, determine the sign of the leading coefficient the x-intercepts the equation for the polynomial function 4. Explain why odd-degree polynomial functions must have at least one x-intercept. 5. Explain why even-degree polynomial functions must have either a maximum or a minimum. 6. i) What is the greatest possible number of zeros each function could have? ii) State the end behaviour for each function. What information did you use to determine this? a) f ( x) x 6x 4x b) g( x) 7x x 8 c) h( x) 4x x x 4 d) p( x) 5x x 7x 5 4 e) f ( x) x x x 6 4 f) h( x) x x 9 7. Consider the polynomial function y = x + x 4x + 5. a) State the degree of the function. b) State the sign of the leading coefficient. c) State the end behaviour of this function. 7
8. Consider the polynomial function y = -x 4 + x. a) State the degree of the function. b) State the sign of the leading coefficient. c) State the end behaviour of this function. 9. The zeros of a quartic function are -, -, and (multiplicity ). a) Determine equations for two functions that satisfy this condition. b) Determine the equation of the function that satisfies this condition and passes through the point (, 4). Answers UL4. a) i) -4, -, and ii) positive for -4 < x < - and x >, negative for x < -4 and - < x < b) i) - and 4 ii) negative for all values of x, x -, 4 c) i) - and ii) positive for x < - and x >, negative for - < x < d) i) - and ii) negative for - < x < and x >, positive for x < -. a) B b) D c) C d) A. a) positive leading coefficient, x-intercepts: - and, positive for - < x < and x >, negative for x < -, y = (x + )(x - ) b) negative leading coefficient, x-intercepts: -4, -, and, positive for x < -4 and - < x <, negative for -4 < x < - and x >, y = -(x + 4)(x + )(x - ) c) negative leading coefficient, x-intercepts: -, -,, and, positive for - < x < - and < x <, negative for x < - and - < x < and x >, y = -(x + )(x + )(x - )(x - ) d) positive leading coefficient, x-intercepts: -,, and, positive for x < - and < x < and x >, negative for - < x <, y = (x + )(x - )(x - ) 4. & 5. Answers will vary 6.a) i) ii) as x, y -, as x -, y b) i) ii) as x, y, as x -, y c) i) ii) as x, y -, as x -, y d) i) 4 ii) x, y -, as x -, y - e)i) 5 ii) as x, y, as x -, y - f)i) 4 ii) as x, y, as x -, y 7.a) rd (cubic) b) + c) as x, y, as x -, y - 8.a) 4 th (quartic) b) - c) as x, y -, as x -, y - 9.a) a) Examples: y = (x + )(x + )(x - ) and y = -(x + )(x + )(x - ) b) y = (x + )(x + )(x - ) 8
UL5 - Solving Polynomials with Technology. Find all rational roots a) x + x 5x + = 0 b) x 4 + x 4 = 0 c) x 4 x 8 = 0 d) x + 8 = 0 e) x 4x + x 8 = 0 f) x 5 5x 4 + 6x 0x 6x + 80 = 0 g) x 5 + x 4 4x 8x x 4 = 0 h)x 5 + x 4 9x 8x + 4x + 8 = 0 i) x 4 49 = 0 Answers UL5.a) mult., - b) none c) ± d) - e) 4 f) 5 g) - h) - i) none manual in the proportions where the width is 5 cm greater than the length. To accommodate the packaging the height of the box must be 0 cm greater than the length. The volume of the box must be 500 cm. What will the dimensions be? 4. An oil tank is being drained. The volume, V(t), in litres, of oil remaining in the tank after t minutes can be modeled by the function V(t) = 0.8(8 t). a) What is the sign of the leading coefficient for this function? (Be careful!) b) How much oil was in the tank initially? c) How much oil was in the tank after 0 minutes? d) After how many minutes was the tank empty? e) How can you determine the domain of the function, in this context, by simply examining the function? UL6 - Applications of Polynomials. The dimensions of a rectangular prism are x-, x-, and x+. The volume of the prism is 4 cm. Find the dimensions of the prism.. A toothpaste box has square ends. The length is cm greater than the width. The volume of the box is 5cm. What are the dimensions of the box? 5. By analyzing the impact of growing economic conditions, a demographer establishes that the predicted population P, of a town t years from now can be modeled by the function P(t) = 6t 4 5t + 00t + 000 a) State the domain of the function in the context of the problem. b) What is the current population of the town? c) What will the population of the town be 0 years from now?. A box holds two CD-ROMs and the instructional manual for a multimedia presentation on why the Dallas Cowboys are the greatest football team EVER! The publishing company can only print the 9
6. The forces acting on a horizontal support beam in a garage cause it to sag by d centimetres, x metres from one end of the beam. The relationship between d and x can be represented by the polynomial 4 d( x) (950x x x ) function 800. a) Graph the polynomial function using technology. b) Determine the maximum deflection (value of d) of the beam. Answers UL6. 7 cm x cm x cm. 5 cm x cm x cm. 5 cm x 0 cm x 5 cm 4.a) negative b) 646.56L c) 6L d) 8 minutes e) The time will start at t = 0 and continue to the first positive x-intercept, the domain is (tεr, 0 t 8}. 5.a) D={t 0} R={d 000} b) 000 c) 69000 6.b) 5.5 0