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B.What is the maximum value of $ k $ if the polynomial $ {x^2} - 3x + k $ can be written as the product of two first degree polynomials?

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A.The polynomial is given as $ {x^2} - 3x + 2 $ .

As the given polynomial $ {x^2} - 3x + 2 $ is a quadratic polynomial. So, factorize the polynomial by breaking the $ - 3x $ term in two parts such that their coefficients multiplication is equal to $ 2 $ and addition is equal to $ - 3 $ .

So, factorize the given polynomial as shown below:

$

\Rightarrow {x^2} - 3x + 2 = {x^2} - x - 2x + 2 \\

= x\left( {x - 1} \right) - 2\left( {x - 1} \right) \\

= \left( {x - 2} \right)\left( {x - 1} \right) \;

$

So, the polynomial $ {x^2} - 3x + 2 $ can be written as $ \left( {x - 2} \right)\left( {x - 1} \right) $ in the form of product of two first degree polynomials.

B.Now in the second part factorize the given polynomial:

$

\Rightarrow {x^2} - 3x + k = {x^2} - 2x - x + k \\

= x\left( {x - 2} \right) - 1\left( {x - k} \right) \;

$

So, this polynomial can only be further written as product of two first degree polynomials if $ x - 2 = x - k $

$\Rightarrow k = 2 $

So, the maximum value of $ k $ for which the polynomial $ {x^2} - 3x + k $ can be written as a product of two first degree polynomials is equal to $ 2 $ .