More about polynomials Examples

ExamplesLet 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

ExamplesWhen 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 \((x  2)(x  3)\). 
SolutionsLet \(Q(x)\) 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 \(4x  3\).

ExamplesThe 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?

SolutionsSuggested 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 \((x  2)(x + 3)\).