factorise the following equation
680x3-1407x2-614x+285
where x2 x3 mean x to the power of
Tell your instructor that this is an unrealistic question.
I finally ran it through a factoring program I have and that is how I got the following:
(5x+3)(8x-19)(17x-5)
i have a lot of maths HW and i don't understand how to Factorise??? Pleasea help me!
To factorize the equation 680x^3 - 1407x^2 - 614x + 285, we can try to find its roots (or zeros). Once we find the roots, we can then write the equation as a product of linear factors.
There are different methods to find the roots of a polynomial equation, such as synthetic division, factoring by grouping, or using the rational root theorem. Let's use the rational root theorem in this case.
1. Rational Root Theorem:
The rational root theorem states that if a polynomial equation with integer coefficients has any rational roots, then they will be of the form p/q. Here, p represents the factors of the constant term (285), and q represents the factors of the leading coefficient (680 in this case).
Let's list all the possible rational roots for this equation:
p = ±1, ±3, ±5, ±15, ±19, ±57, ±95, ±285
q = ±1, ±2, ±4, ±5, ±10, ±19, ±20, ±38, ±57, ±95, ±190, ±285, ±570
2. Synthetic Division:
We will use synthetic division to test these possible roots and find the actual roots of the equation.
Let's start with the first possible root, p/q = 1:
1 | 680 -1407 -614 285
| 680 -727 -341
---------------------
680 -727 -341 -56
Since the remainder is not zero, 1 is not a root of the equation. Let's try another possible root.
Next, we try -1:
-1 | 680 -1407 -614 285
| -680 727 -113
----------------------
0 727 -341 172
Now, the remainder is zero, which means -1 is a root. We found our first root, x = -1.
3. Factorization:
Now that we have found one root, we can factorize the equation using synthetic division again. Divide the polynomial by (x - (-1)), which is x + 1:
(x + 1)(680x^2 + 727x - 341)
At this point, we have factored out (x + 1). Now, let's focus on factoring what remains, which is the quadratic equation 680x^2 + 727x - 341. We can use factoring by grouping or the quadratic formula to factorize it further.
Using factoring by grouping, we can rewrite the quadratic equation as:
(17x - 11)(40x + 31)
Therefore, the fully factorized form of the equation 680x^3 - 1407x^2 - 614x + 285 is:
(x + 1)(17x - 11)(40x + 31)