195x^5 + 168x^4 - 28x^3 - 24x^2
This is factoring I have looked at everything.
I need help working it out. This is poloynomials.
Well, let's look at everything again.
clearly x^2 is a common factor.
Look at the numbers, start with the 24
24 = 3*8 = 3*2*2*2
so the 195 is only divisible by 3 but not 28, so
the HCF is x^2
195x^5 + 168x^4 - 28x^3 - 24x^2
= x^2(195x^3 + 168x^2 - 28x - 24)
The x^2 is easy:
195x^5 + 168x^4 - 28x^3 - 24x^2
x^2(195x^3+168x^2-28x-24)
Now things get tough. If there are any rational roots of the form p/q, then we know that
p divides 24
q divides 195
Some time and effort should convince you that there are no other rational roots, so no factors using integers.
To factor the expression 195x^5 + 168x^4 - 28x^3 - 24x^2, we can begin by looking for common factors among the terms. In this case, the only common factor is 1, so we can't factor out a common factor.
Next, we can try factoring by grouping. To do this, we group the terms in pairs and look for common factors within each pair:
(195x^5 + 168x^4) - (28x^3 + 24x^2)
Within the first pair, we can observe that both terms have a common factor of x^4. Factoring out x^4, we have:
x^4(195x + 168) - (28x^3 + 24x^2)
Now, within the second pair, we can observe that both terms have a common factor of 4x^2. Factoring out 4x^2, we have:
x^4(195x + 168) - 4x^2(7x + 6)
After factoring out the common factors, we can see that we have a common binomial factor of (195x + 168). Factoring out this common binomial factor, we get:
(195x + 168)(x^4 - 4x^2(7x + 6))
Therefore, the factored form of the expression 195x^5 + 168x^4 - 28x^3 - 24x^2 is (195x + 168)(x^4 - 4x^2(7x + 6)).