Exercises 9 (Long division and the Remainder theorem)

  1. Show that x3 is a factor of 6x319x2+x+6 and hence determine the other factors.

  2. Given that x=0.25 is one root which satisfies the equation 4x3+3x25x+1=0. Determine the other roots.

  3. Establish the remainder when:

    1. x4 is divided into 24x45x3+3x
    2. x+5 is divided into 4x3+25x2+20x25
    3. x2 is divided into 12x3+x238x24
  4. Determine the quotient and remainder when:

    1. x2 is divided into x3+14x2+56x+14
    2. x+2 is divided into 4x4+3x221x+4
  5. Solve for x: 2x35x2x+6=0.

  6. The power of a flat belt is given by the formula P=Av+Bv3, where v is the linear velocity of the belt and A and B are constants. Given that A=12 and B=1, determine the value of the velocity when P=16.

  7. The value of the magnetic field at a given point due to the presence of two magnets of equal moment, M, is given by F=2Md312Md32, where d1 and d2 are the distances of the magnets from the given point. Show that the field may be expressed as F=2M((d2d1)(d22+d1d2+d21)d31d32). (Hint: Starting with the equation (1), add the fractions and then find a factor for the numerator.)

  8. Use algebraic long division to find the quotient and the remainder for the following:

    1. x^3 + 2x^2 – x – 2 divided by x – 1
    2. 2x^3 + 9x^2 – 4x – 21 divided by 2x – 3
    3. x^4 + x^3 + 7x – 3 divided by x^2 – x + 3
    4. 6x^4 + 14x^3 – 9x^2 – 7x + 3 divided by 2x^2 – 1
  9. Find the quotient and remainder for:

    1. \dfrac{x^2+6x-2}{x^2+4x+1}
    2. \dfrac{2x^2+5}{x^2+1}
    3. \dfrac{5x^2+2x-11}{x^2+x-2}
    4. \dfrac{x^3-5x^2+9x-7}{x^2-2x+8}
  10. Use the remainder theorem to find the remainder when:

    1. 6x^3 + 7x^2 - 15x + 4 is divided by (x - 1)
    2. 2x^3 - 3x^2 + 5x + 4 is divided by (x + 1)
    3. x^3 - 7x^2 + 6x + 1 is divided by (x - 3)
    4. 5 + 6x + 7x^2 - x^3 is divided by (x + 2)
    5. x^4 - 3x^3 + 2x^2 + 5 is divided by (x - 1)
  11. Factorise:

    1. x^3 - 2x^2 - 5x + 6
    2. 2x^3 + 7x^2 - 7x - 12
    3. 2x^3 + 3x^2 - 17x + 12
    4. 6x^3 - 5x^2 - 17x + 6
    5. 2x^4 + 7x^3 - 17x^2 -7x + 15
    6. 6x^4 + 31x^3 + 57x^2 + 44x + 12
  12. Given that x + 2 is a factor of 2x^3 + 6x^2 + bx - 5, find the value of b and then find the remainder when the expression is divided by (2x - 1).

  13. The expression 3x^3 + 2x^2 - bx + a is exactly divisible by (x - 1), but leaves a remainder of 10 when divided by (x + 1). Find the values of a and b.

  14. The expression 8x^3 - 4x^2 + ax + b gives a remainder of -19 when divided by (x + 1) and a remainder of 2 when divided by (2x - 1). Find the values of a and b.

Solutions: 1. (x-3)(3x-2)(2x+1); 2. -1.618, 0.618; 3. (i) 5836; (ii) 0; (iii) 0; 4. (i) Q=x^2+16x+88, r=190; (ii) Q=4x^3-8x^2+19x-59, r=122; 5. 2,3/2,-1; 6. v=2 (twice), or v=-4; 8. (a) x^2+3x+2 rem 0; (b) x^2+6x+7 rem 0; (c) x^2+2x-1 rem 0; (d) 3x^2+7x-3 rem 0; 9. (a) 1 rem 2x-3; (b) 2 rem 3; (c) 5 rem -3x-1; (d) x-3 rem -5x+17; 10. (a) 2; (b) -6; (c) -17; (d) 29; (e) 5; 11. (a) (x -1)(x - 3)(x + 2); (b) (2x - 3)(x + 4)(x + 1); (c) (2x - 3)(x + 4)(x - 1); (d) (3x - 1)(2x + 3)(x - 2); (e) (x + 1)(x - 1)(2x - 3)(x + 5); (f) (x + 1)(x + 2)(3x + 2)(2x + 3); 12. b = 3/2 and the reminder when divided by (2x-1) is -5/2; 13. a=3, b=8; 14. a=6,b=-1