Exercises 9 (Long division and the Remainder theorem)

  1. Show that \(x - 3\) is a factor of \(6x^3 - 19x^2 + x + 6\) and hence determine the other factors.

  2. Given that \(x = 0.25\) is one root which satisfies the equation \(4x^3 + 3x^2 - 5x + 1 = 0\). Determine the other roots.

  3. Establish the remainder when:

    1. \(x - 4\) is divided into \(24x^4 - 5x^3 + 3x\)
    2. \(x + 5\) is divided into \(4x^3 + 25x^2 + 20x - 25\)
    3. \(x - 2\) is divided into \(12x^3 + x^2 - 38x - 24\)

  4. Determine the quotient and remainder when:

    1. \(x - 2\) is divided into \(x^3 + 14x^2 + 56x + 14\)
    2. \(x + 2\) is divided into \(4x^4 + 3x^2 - 21x + 4\)

  5. Solve for \(x\): \(2x^3-5x^2-x+6=0\).

  6. The power of a flat belt is given by the formula \(P = Av + Bv^3\), 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 \[F=\frac{2M}{d_1^3}-\frac{2M}{d_2^3}, \tag{1}\] where \(d_1\) and \(d_2\) are the distances of the magnets from the given point. Show that the field may be expressed as \[F=2M\left(\frac{(d_2-d_1)(d_2^2+d_1d_2+d_1^2)}{d_1^3d_2^3}\right). \tag{2}\] (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. \(a=6,b=-1\)