EXERCISES AND PROBLEMS

50 Find the values that a and b must have so that when we divide P (x) = x5 + ax 4 + x 2 + bx + 8 by (x + 1) the remainder is 9, and when we divide it by (x – 2) the remainder is 6.

51 Find the lowest common multiple and the greatest common divisor for each of the following polynomials: a) x 2 ; x 2 – x; x 2 – 1 b) x – 3; x 2 – 9; x 2 – 6x + 9 c) 2x; 2x + 1; 4x 2 – 1 d) x + 2; 3x + 6; x 2 + x – 2

52 Calculate the GCD and LCM of the following polynomials from exercise 30: a) Sections a and c. b) Sections b and c.

53 Check whether there is any divisibility relationship between these pairs of polynomials: a) P (x) = x 4 – 4x 2 y Q (x) = x 2 – 2x b) P (x) = x 2 – 10x + 25 y Q (x) = x 2 – 5x c) P (x) = x 3 + x 2 – 12x y Q (x) = x – 3

54 Factorise the following expressions:

Advanced Problem Solving

59 Inside rectangle ABCD with sides AB = 3 cm and BC = 5 cm, we have inscribed the quadrilateral '' ABCD'' where '' '' AA BB CC DD == = = x. Write the area of '' ABCD'' as a function of x

55 Simplify the following algebraic fractions:

60 In the triangle below, BC = 10 cm, AH = 4 cm. We draw a point D on the line so that AD = x and we draw a line MN through D, parallel to BC Starting at points M and N we draw lines that are perpendicular to BC a) Express MN as a function of x. (Use the similarity of triangles AMN and ABC). b) Write a polynomial expressing the area of rectangle MNPQ as a function of x.

61 We have a rectangle with a perimeter of 20 cm. If the base is reduced by 2 cm and the height by 3 cm, by how much is the area of the rectangle reduced? Express it as a function of the base.

62 A triangle has a base of 20 cm and a height of 15 cm. If the height is increased by x % and the base by (x + 2) %, express the new area of the triangle as a function of x.

56 A tap takes 4 hours to fill a tank. Another tap takes 6 hours to fill the same tank. If both taps are open and the plug is taken out, it takes 12 hours to fill the tank. Express, as a function of x, how long it takes to empty the tank if both taps are closed.

57 Express the total surface area of this truncated pyramid as a function of x: x + 1 is the height of a lateral face.

58 M is any point along segment AB which is 6 cm. Express the blue shaded area as a function of xAM = .

63 A shop owner sold two bicycles. He made a 20% profit on one of the bicycles, and a loss of 10% on the other one. In total, he made a profit of 15%. Use algebra to express this statement.

64 We cut a piece of wire that is 1 m long into two unequal parts. With one of these parts, we form an equilateral triangle, and with the other, a square. Write the sum of the areas of both figures.

65 Using a rectangular piece of card that measures 30 cm and 20 cm, we make a box without a lid by cutting a square with sides of length each corner. Write the volume of the box as a function of x.

You Can Also Try This

66 a) Check that ) nn nn 1 1 1 1 1

( + = + b) What will the value of A be in the following expression?

·· · · A 12 1 23 1 34 1 910 1 = +++ +

And if the last summand were · 99 100 1 ?

67 Prove that when a second-degree polynomial is divided Px ax bx c () 2 =+ + by Qx x 1 – () = , the remainder of the division is equal to the sum of the P coefficients (x).

Would this be true for ()Px ax bx cx d 32 =+ ++ ?

UNDERSTAND AND APPLY IN THE CHALLENGE

68 How many squares do stairs with 5 steps have? Write a formula to show how many squares there are in stairs with n steps.

69 If we increase the length of the base of a rectangle by x % and its height by 2x %, the area increases by 31 %. Write this in algebraic language.

70 Find a third-degree polynomial that we know is divisible by (2x 2 – 3), the remainder of dividing it by (x + 1) is 15 and whose highest degree coefficient is 6.

Remember The Theory Reflect

71 What do the values of a and b have to be for the polynomials P (x) and Q (x) to be equal?

P (x) = x 3 – (4 + a)x + (1 + b ) Q (x) = (a + 3)x 3 + (a + 2)x 2 – 2x + 5 a) Write three first-degree divisors of P (x). b) Write one second-degree divisor of P (x). b) If –5 is a root of the polynomial P (x), what can you say about the division P (x) : (x + 5)? c) What result did you use to answer these two questions? a) If a polynomial has a degree of 3, and another polynomial has a degree of 2, their product will have a degree of 6. b) If P (0) = 1, then P (x) is divisible by (x – 1). c) If we add two third-degree polynomials together, we always get a third-degree polynomial. d) If P (3) ≠ 0, the polynomial P (x) is not divisible by x – 3. e) If P (–2) = 0, then x + 2 is a factor of P (x). f ) If P (x) = ax 2 + bx + 2 y P (±2) ≠ 0, then P (x) cannot have integer roots. g) It is not possible to write a fourth-degree polynomial that only has a triple root. h) A polynomial that is zero for the numbers 0, 1, 2, 3, 4 cannot be a fourth-degree polynomial.

72 The roots of P (x) are 0, 2 and –3.

73 a) If the division P(x) : (x – 2) P (x) : (x – 2) is exact, what can you say about the value P (2)?

74 True or false? Explain your answer and give examples.

75 Invent two second-degree polynomials that meet the condition given in each case: a) LCM [P (x), Q (x)] = x 2(x – 3)(x + 2) b) GCD [P (x), Q (x)] = 2x + 1

76 Given the polynomial P (x) = (x – 1)2(x + 3) find a second-degree polynomial, Q(x), Q (x), that meets the following conditions: a) GCD [P (x), Q (x)] = x – 1 b) LCM [P (x), Q (x)] = (x – 1)2(x 2 – 9)

77 By which fraction do we need to multiply x x 1 5 ––to get xx xx 34 5 ––2 2 + ?

78 Prove that the polynomial x2 + (a + b)x + ab is divisible by x + a and by x + b or any value of a and b. What would they look like broken down into factors?

79 If a polynomial A has a degree of 4 and another polynomial B has a degree of 3, state the degree of the following polynomials: a) Sum of A and B. b) Subtraction of A and B. c) Product of A and B. d) Quotient of A and B a) When three polynomials with different degrees are added together, the resulting polynomial has the same degree as the highest of them. b) The result of adding three third-degree polynomials is a sixth-degree polynomial. c) The result of adding two fifth-degree polynomials is another polynomial with a degree of 5 or higher.

80 True or false?