Solve the following:
(a) Write the polynomial $x^{2} - 3 x + 2$ as a product of two first degree polynomials.
(b) What is the maximum value of $k$ if the polynomial $x^{2} - 3 x + k$ can be written as the product of two first degree polynomials.

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Solution

(a) $x^{2} - 3 x + 2 = x^{2} - 2 x - x + 2 = x (x - 2) - 1 (x - 2) = (x - 1) (x - 2)$

(b) Since $x^{2} - 3 x + k$ can be written as the product of two first degree polynomials,
The maximum value of $k$ is given by,
$b^{2} - 4 a c = 0$
$\Rightarrow {(- 3)}^{2} - 4 (1) (k) = 0$
$\Rightarrow 9 - 4 k = 0$
$\Rightarrow k = 9 / 4$

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