• 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 $$f(x) = 1$$.
• (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

• 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 $$(x - 2)(x - 3)$$.
• 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 $$4x - 3$$.
• 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?
$$\begin{array}{1}{x^2} - 4 = (x - 2)(x + 2)\\{x^2} - 5x + 6 = (x - 2)(x - 3)\end{array}$$
The remaining polynomial may be $$(x - 2)(x + 3)$$.