Question

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

Answer 1

Answer:

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

Step-by-step 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) $a = 3$

$b = 7 - a$

$b = 4$

$a\cdot b = k$

$k = (3)\cdot (4)$

$k = 12$