for what integral value(s) of K can the trinomial x^2 + kx +12 be factored?

my English is limited, I not sure what the question is asking, can everyone help me?

Thank You!

integral values are integer numbers, so k cannot be a fraction

possible factors that end in +12:

(x+1)(x+12)
(x-1)(x-12)
(x+2)(x+6)
(x-2)(x-6)--> x^2 - 8x + 12 , so k = -8
(x+3)(x+4)
(x-3)(x-4)

I expanded one of them, to show one value of k
You do the others

Why did the math book go to therapy?

Because it had too many problems to factor in its life!

In order for the trinomial x^2 + kx + 12 to be factored, we need to find integral values of K. One way to approach this is to factor the trinomial completely by finding two binomials whose product is x^2 + kx + 12. This can be done by using the fact that the product of two binomials is equal to the trinomial. So let's try to factor it out:

We need to find two integers A and B such that (x + A) and (x + B) when multiplied, give us x^2 + kx + 12.

Expanding (x + A)(x + B) gives us x^2 + (A + B)x + (AB).

So A + B should be equal to k, and AB should be equal to 12. By finding pairs of integers that satisfy these conditions, we can determine the possible values of K.

For example, if we take A = 3 and B = 4, we get (x + 3)(x + 4) = x^2 + 7x + 12. Here, k is equal to 7.

Similarly, if we take A = -2 and B = -6, we get (x - 2)(x - 6) = x^2 - 8x + 12. Here, k is equal to -8.

So, for K to have integral values, it can take on the values 7 and -8.

Of course! Let's break down the question.

The trinomial given is x^2 + kx + 12.

The question is asking for what integral values of K can this trinomial be factored. In other words, we need to find the values of K where the trinomial can be written as the product of two binomials.

To determine this, we need to find the factors of 12 that add up to K.

The factors of 12 are:
1, 2, 3, 4, 6, 12

Now, let's check which combinations of these factors add up to K.
For example, if we choose 1 as a factor, the other factor needs to be K-1. Similarly, if we choose 2 as a factor, the other factor needs to be K-2, and so on.

Let's list down the possible combinations:

1 (K-1)
2 (K-2)
3 (K-3)
4 (K-4)
6 (K-6)
12 (K-12)

To factor the trinomial, we need to find the values of K where these combinations can be simplified further.

If we simplify each combination:
1 (K-1) = K - 1
2 (K-2) = 2K - 4
3 (K-3) = 3K - 9
4 (K-4) = 4K - 16
6 (K-6) = 6K - 36
12 (K-12) = 12K - 144

Now, we need to find the values of K that make these combinations factorable. In other words, the factors should have a common term.

Let's analyze each combination:
K - 1: This combination is factorable for any integral value of K.
2K - 4: This combination can be factored if the common factor is 2.
3K - 9: This combination can be factored if the common factor is 3.
4K - 16: This combination can be factored if the common factor is 4.
6K - 36: This combination can be factored if the common factor is 6.
12K - 144: This combination can be factored if the common factor is 12.

So, the trinomial x^2 + kx + 12 can be factored for the following integral values of K: -1, 2, 3, 4, 6, 12.

I hope that helps! Let me know if you have any further questions.

Of course! Your question is asking for what integral values of K can the trinomial x^2 + kx + 12 be factored. To solve this, we need to determine the factors of the quadratic trinomial.

A quadratic trinomial can be factored if it can be expressed as the product of two binomials. In this case, the trinomial can be factored as (x + a)(x + b), where a and b are the unknown values we are trying to find.

Expanding the product (x + a)(x + b) gives us x^2 + (a+b)x + ab. We can equate the coefficients of the trinomial x^2 + kx + 12 with the expanded form, which gives us the following equations:

a + b = k (1)
ab = 12 (2)

To find integral values for a and b, we need to consider the factors of 12. The factors of 12 are: 1, 2, 3, 4, 6, and 12. We can now check each pair of factors and see if they satisfy equation (1).

For example, if we try a = 1 and b = 12, we would have 1 + 12 = 13, which is not equal to k. We can continue this process and see if any pair of factors satisfies equation (1). If we find such a pair, we have found the integral values of K that make the trinomial factorable.

In this case, if we try a = 2 and b = 6, we have 2 + 6 = 8, which is equal to k. Therefore, the quadratic trinomial x^2 + kx + 12 can be factored when k = 8.

So, the integral value of K that makes the trinomial factorable is K = 8.