Solving Polynomials: Factoring x^3-2x^2-x+2

• mbrmbrg
In summary, the conversation discusses finding the expression x^3-2x^2-x+2 in the form of (x-1)(x+1)(x-2). The Rational Root Theorem is mentioned as a possible method for finding the roots of the polynomial, and factoring by grouping is also suggested as another approach.
mbrmbrg
I'm going over some math work, and ran across the following:
$$\frac{2x^2-5x-1}{x^3-2x^2-x+2}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x-2}$$

How do you get $$x^3-2x-x+2=(x-1)(x+1)(x-2)$$?

Well, it's easy to verify that it's right.

To find it... well, one way is that there are only four rational numbers that could possibly be roots of your polynomial, so you just try them!

Hurkyl said:
To find it... well, one way is that there are only four rational numbers that could possibly be roots of your polynomial
How so? (pardon me if I be asking heap stupid question; I pulled an all-nighter last night and the brain doth rebel against unwarranted abuse)

Why do I feel like someone at PhysicsForums sent me to this webpage before...?

Thanks!

Sometimes, especially for third degree expressions, you can see if you are able to factor by grouping. Namely,

$$x^3 - 2x^2 - x + 2 = (x^3 - 2x^2) + (-x + 2) = x^2(x - 2) - (x - 2) = (x - 2)(x^2 - 1) = (x - 2)(x - 1)(x + 1)$$

1. What is the process for factoring a polynomial like x^3-2x^2-x+2?

To factor a polynomial, we need to find its factors, which are numbers or expressions that when multiplied together, give the original polynomial. In the case of x^3-2x^2-x+2, we can use the grouping method to factor it. This involves grouping the terms in pairs and factoring out a common factor from each pair. Then, we can factor out another common factor from the remaining terms to get the final factored form.

2. How do I know if a polynomial can be factored?

A polynomial can be factored if it has more than one term and all its terms have a common factor. In other words, if all the terms can be divided by the same number or expression, then the polynomial can be factored.

3. Can we use the quadratic formula to factor a polynomial like x^3-2x^2-x+2?

No, the quadratic formula can only be used to solve quadratic equations of the form ax^2+bx+c=0. It cannot be used to factor polynomials with a degree higher than 2.

4. Are there any shortcuts or tricks for factoring polynomials?

Yes, there are some common patterns that can help with factoring polynomials. These include the difference of squares, perfect square trinomials, and the sum/difference of cubes. These patterns can save time and make factoring easier.

5. How can factoring polynomials be useful in real life?

Factoring polynomials is a fundamental skill in algebra and is used in various applications such as solving equations, finding roots, and simplifying expressions. In real life, it can be used in fields like engineering, finance, and science to model and solve problems involving variables and equations.

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