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A.Write the polynomial x23x+2 as a product of two first degree polynomials.
B.What is the maximum value of k if the polynomial x23x+k can be written as the product of two first degree polynomials?

Answer
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Hint: In the first part factorize the given quadratic polynomial to write it as product of two first degree polynomials and in the second part factorize the polynomial and check whether for what value of k the equation can be factorized.

Complete step-by-step answer:
A.The polynomial is given as x23x+2 .
As the given polynomial x23x+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:
  x23x+2=x2x2x+2=x(x1)2(x1)=(x2)(x1)
So, the polynomial x23x+2 can be written as (x2)(x1) in the form of product of two first degree polynomials.
So, the correct answer is “ (x2)(x1) ”.

B.Now in the second part factorize the given polynomial:
 x23x+k=x22xx+k=x(x2)1(xk)
So, this polynomial can only be further written as product of two first degree polynomials if x2=xk
k=2
So, the maximum value of k for which the polynomial x23x+k can be written as a product of two first degree polynomials is equal to 2 .
So, the correct answer is “k=2”.

Note: In the first part the polynomial is given and need to write it as a product of two first degree polynomials but factorizing it. In the next part a polynomial contains a value as a constant which can only be found with the given condition that the polynomial can be written as a product of two first degree polynomials.