help my daughter has a math problem and came to me for help, I am lost....
what is the gcf for the following trinomial?
x^3-18x^2+28x
I am assumming you are trying to factorise
we can simply take out an x to give:
x(x^2-18x+28)
the quadratic will not factorise simply but if equated to zero would sove to give two real roots.
Asking for the gcf implies that there ought to be at least two things that you're trying to factorize (because the gcf is the "greatest common factor"), whereas here there's only one. I think part of the question must have been lost somewhere.
To find the greatest common factor (GCF) of the given trinomial, x^3 - 18x^2 + 28x, we can factor out an x from each term:
x(x^2 - 18x + 28)
Now we have a quadratic expression, x^2 - 18x + 28, which we will further factorize.
To factorize the quadratic, we need to find two numbers that, when multiplied, give us 28, and when added, give us -18 (the coefficient of the linear term).
The factors of 28 are: 1, 2, 4, 7, 14, 28
We need to find two numbers whose sum is -18. From the factors above, we can see that -2 and -14 fit the criteria because -2 + (-14) = -16.
So, we rewrite the quadratic expression by splitting the middle term:
x(x^2 - 2x - 14x + 28)
Now, we group the terms:
x[(x^2 - 2x) + (-14x + 28)]
And now we factor by grouping:
x[x(x - 2) - 14(x - 2)]
We can see that we have a common binomial factor, (x - 2), in both terms.
Therefore, we can factorize the trinomial as:
x(x - 2)(x - 14)
So, the greatest common factor (GCF) of the given trinomial x^3 - 18x^2 + 28x is x(x - 2)(x - 14).