Find all positive values for k for which each of the following can be factored.
I have 2 problems i think i got the one right but not sure the second one i have no idea where to even start
2x^3+16x^2-40x=2x(x^2+8x-20)
(x-2)(x+10)=8x
this one i am stumped
can some one explain to me how i do this
x^2+x-k
Well, the first one is odd. In the first line, it is correct. Then, the final factor should be:
2x(x-2)(x+10)
I don't know where you got 8x.
The second. Find all positive values of k....
x^2+x-k
Looking at the quadratic equation, the serd b^2-4ac, leads to 1+4k. For the square root of this to be real,
1+4k>=0
or k>=-1/4 however, k has to be a positive number, so k>0 leads to solutions. Any value of k >0 lets it be factorable.
Here is a comment by a great tutor to me:
Like you said, the discriminant is 1+4k
However, for a polynomial to be “factored” it is generally agreed that the factors are to contain rational numbers.
For that to happen the 1+4k must be a perfect square
Values of k which would work are 2,6,12,20,30,42,….we are really just looking for a number that has factors that differ by one.
This sequence can be obtained by the quadratic n^2 + n , where n is a set of the natural numbers.
e.g. if n=57, then k=57^2 + 57 = 3306
x^2 + x – 3306
=(x+58)(x-57)
To find the positive values of k for which the quadratic expression x^2 + x - k can be factored, we need to consider the discriminant. The discriminant is the expression inside the square root in the quadratic formula, which is b^2 - 4ac.
In this case, the quadratic expression is in the form ax^2 + bx + c.
For x^2 + x - k, we have a = 1, b = 1, and c = -k.
So the discriminant is 1^2 - 4(1)(-k) = 1 + 4k.
For the quadratic expression to be factorable over the rational numbers, the discriminant must be a perfect square. In other words, 1 + 4k must be a perfect square.
To find the positive values of k, we need to find the values for which 1 + 4k is a perfect square.
A perfect square is a number that can be written as the square of an integer. So we need to find values of k such that 1 + 4k is the square of an integer.
If we think about the perfect squares, we can notice that the difference between consecutive perfect squares is always the same. For example, the difference between 1^2 and 2^2 is 3, the difference between 2^2 and 3^2 is 5, and so on.
So if we find a number that is a perfect square and subtract the previous perfect square, we should get a value of 4. This means that 1 + 4k = 4, and solving for k, we get k = 1/4.
But the question asks for positive values of k, so we need to find values of k greater than 1/4.
By using this pattern, we can find that the values of k that work are 2, 6, 12, 20, 30, 42, and so on. These values can be obtained using the formula n^2 + n, where n is a set of natural numbers. For example, if n = 2, then k = 2^2 + 2 = 6. If n = 3, then k = 3^2 + 3 = 12.
So the positive values of k for which x^2 + x - k can be factored are 2, 6, 12, 20, 30, 42, and so on. To factor the quadratic expression, you can use the factors obtained using the formula (x + n)(x - (n-1)). For example, if k = 6, then the factored expression is (x + 3)(x - 2).