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Find two values for k such that the trinomial x2 – 7x + k can be factored over the integers. Explain your reasoning.

1 Answer

4 votes

Answer:

1) For
a = 1:
b = 6 and
k = 6, 2) For
a = 3:
b = 4 and
k = 12

Explanation:

The polynomial
y = x^(2) - 7\cdot x + k is a second-order polynomial of the form
(x-a)\cdot (x-b) = x^(2)-(a+b)\cdot x + a\cdot b. By direct comparison, we construct the following system of equations:


a + b = 7 (1)


a\cdot b = k (2)

By (1) we know that there are a family of pairs such that the system of equations is satisfied. Let suppose that both
a and
b are integers. We assume two arbitrary integers for
a:

1)
a = 1


b = 7 - a


b = 6


a\cdot b = k


k = (6)\cdot (1)


k = 6

2)
a = 3


b = 7 - a


b = 4


a\cdot b = k


k = (3)\cdot (4)


k = 12

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