Answer:
1) For [tex]a = 1[/tex]: [tex]b = 6[/tex] and [tex]k = 6[/tex], 2) For [tex]a = 3[/tex]: [tex]b = 4[/tex] and [tex]k = 12[/tex]
Step-by-step explanation:
The polynomial [tex]y = x^{2} - 7\cdot x + k[/tex] is a second-order polynomial of the form [tex](x-a)\cdot (x-b) = x^{2}-(a+b)\cdot x + a\cdot b[/tex]. By direct comparison, we construct the following system of equations:
[tex]a + b = 7[/tex] (1)
[tex]a\cdot b = k[/tex] (2)
By (1) we know that there are a family of pairs such that the system of equations is satisfied. Let suppose that both [tex]a[/tex] and [tex]b[/tex] are integers. We assume two arbitrary integers for [tex]a[/tex]:
1) [tex]a = 1[/tex]
[tex]b = 7 - a[/tex]
[tex]b = 6[/tex]
[tex]a\cdot b = k[/tex]
[tex]k = (6)\cdot (1)[/tex]
[tex]k = 6[/tex]
2) [tex]a = 3[/tex]
[tex]b = 7 - a[/tex]
[tex]b = 4[/tex]
[tex]a\cdot b = k[/tex]
[tex]k = (3)\cdot (4)[/tex]
[tex]k = 12[/tex]
