More about polynomials Examples


  • Examples
    Let p and q be constants. When \(f(x) = 2{x^3} + p{x^2} + qx\) is divided by \(2x + 1\), the quotient is \({x^2} - 1\) and the remainder is 1.
    (a) Find the values of p and q.
    (b) Hence solve the equation \(f(x) = 1\).
  • Solutions

    (a) ∵ Dividend = Quotient x Divisor + Remainder
    ∴ L.H.S.\( = 2{x^3} + p{x^2} + qx\)
    R.H.S.\(\begin{array}{l} = ({x^2} - 1)(2x + 1) + 1\\ = 2{x^3} - 2x + {x^2} - 1 + 1\\ = 2{x^3} + {x^2} - 2x\end{array}\)
    By comparing the like terms on the L.H.S. and R.H.S., we have \(\underline{\underline {p = 1{\rm{ }},{\rm{ }}q = - 2}} \).

    \(\begin{array}{1}(b)f(x) = 1\\2{x^3} + {x^2} - 2x = 1\\({x^2} - 1)(2x + 1) + 1 = 1\\({x^2} - 1)(2x + 1) = 0\\(x + 1)(x - 1)(2x + 1) = 0\end{array}\) [ From the result of (a) ]
    \(x + 1 = 0\) or \(x - 1 = 0\) or \(2x + 1 = 0\)
    \(x = - 1\) or \(x = 1\) or \(x = - \frac{1}{2}\)
    ∴ \(x = - 1{\rm{ }},{\rm{ }} - {\rm{ }}\frac{1}{2}\) or 1


  • Examples
    When a polynomial \(f(x)\) is divided by \(x - 2\) and \(x - 3\), the remainders are 5 and 9 respectively.
    Find the remainder when \(f(x)\) is divided by \((x - 2)(x - 3)\).
  • Solutions
    Let \(Q(x)\) be the quotient and \(cx + d\) be the remainder when \(f(x)\) is divided by \((x - 2)(x - 3)\), where c and d are constants.
    ∴ \(f(x) = Q(x) \times (x - 2)(x - 3) + (cx + d)\)
    ∵ The remainder is 5 when \(f(x)\) is divided by \(x - 2\).
    ∴ \(\begin{array}{c}f(2) = 5\\Q(2) \times (2 - 2)(2 - 3) + [c(2) + d] = 5\\2c + d = 5\;........\;(1)\end{array}\)
    ∵ The remainder is 9 when \(f(x)\) is divided by \(x - 3\).
    ∴ \(\begin{array}{c}f(3) = 9\\Q(3) \times (3 - 2)(3 - 3) + [c(3) + d] = 9\\3c + d = 9\;........\;(2)\end{array}\)
    (2) - (1), \(\begin{array}{c}3c + d - 2c - d = 9 - 5\\c = 4\end{array}\)
    Substitute \(c = 4\) into (1),
    \(\begin{array}{c}2(4) + d = 5\\8 + d = 5\\d = - 3\end{array}\)
    ∴ The remainder is \(4x - 3\).


  • Examples
    The H.C.F. and L.C.M. of three polynomials are \(x - 2\) and \((x - 2)(x + 2)(x - 3)(x + 3)\) respectively. It is given that two of the polynomials are \({x^2} - 4\) and \({x^2} - 5x + 6\), what is the remaining one?
  • Solutions
    Suggested answer:
    \(\begin{array}{1}{x^2} - 4 = (x - 2)(x + 2)\\{x^2} - 5x + 6 = (x - 2)(x - 3)\end{array}\)
    The remaining polynomial may be \((x - 2)(x + 3)\).