Is it possible to factor x^3 - 9x^2 + 24x - 20 and solve for x by hand?

I tried using the grouping method and got x^2(x-9) + 4(6x-5) but how can I continue solving for x?

If I use a calculator I get x= 2 and 5 but is it possible to find this by hand? thanks

You use rational root theorem.

Let p = constant term
Let q = numerical coefficient of highest degree of x
x = +/- p/q

In here, p = -20 and q = 1. Thus,
x = (+/-) 1 , 2 , 4 , 5 , 10, 20

You have to substitute each factor (for instance x = -1) to the expression, and if you get a zero answer, that value of x is a factor. Well, it's kind of like trial and error, but at least you know which ones are to be substituted.

Hope this helps :3

*I mean you get all the factors of x = +/- p/q. In the problem, x = +/- 20, and its factors are written there above. The factors are the ones you have to check by substituting them to the expression. :)

After showing that both f(1) and f(-1) ≠ 0

try f(2)
= 8 -36+48-20 = 0
so (x-2) is a factor
Now use synthetic division and
x^3 - 9x^2 + 24x - 20
= (x-2)(x^2 - 7x + 10)
= (x-2)(x-2)(x-5)

so x^3 - 9x^2 + 24x - 20
= (x-5)(x-2)^2

For a cubic, once you find one root, you can reduce your cubic to a quadratic, by either long or synthetic division.
Of course you could just feel lucky and keep going to find other roots from the list that Jai gave you, and you would hit x = 5

And of course, as soon as you found 2 factors of a cubic you can "reason" out what the third factor would be.

Yes, it is possible to factor x^3 - 9x^2 + 24x - 20 by hand.

You've correctly factored out the common factors from the original expression using the grouping method: x^2(x-9) + 4(6x-5). Now, let's continue solving for x.

We have two terms left: x^2(x-9) and 4(6x-5). To factor x^2(x-9), we can further break it down as follows:

x^2(x-9) = x^2 * x - x^2 * 9 = x^3 - 9x^2

So now our expression becomes: x^3 - 9x^2 + 4(6x-5).

Expanding 4(6x-5), we get: 4 * 6x - 4 * 5 = 24x - 20.

Now our expression becomes: x^3 - 9x^2 + 24x - 20.

At this point, we have fully factored the expression. Now we can set it equal to zero and solve for x:

x^3 - 9x^2 + 24x - 20 = 0

To solve this equation by hand, we can use factoring by grouping or synthetic division. However, in this case, factoring by grouping is not straightforward.

Therefore, let's use the Rational Root Theorem to find the possible rational roots of this polynomial. The Rational Root Theorem states that if a rational number p/q is a root of a polynomial with integer coefficients, then p must be a factor of the constant term (in this case, -20), and q must be a factor of the leading coefficient (in this case, 1).

The factors of 20 are ±1, ±2, ±4, ±5, ±10, and ±20, and the factors of 1 are ±1. So the possible rational roots of the polynomial are: ±1, ±2, ±4, ±5, ±10, and ±20.

We can try these possible rational roots by substituting them into the polynomial expression and checking if they make the polynomial equal to zero. This is a trial and error process.

After trying the possible rational roots, we find that x = 2 and x = 5 make the polynomial equal to zero.

Therefore, the solutions to the equation x^3 - 9x^2 + 24x - 20 = 0 are x = 2 and x = 5.

So, by hand, we have found the values of x without using a calculator.