Factoring Questions: Tips & Hints for Success

[wScott]

Are there any hints in a factoring question that tell you automatically if they can be factored?

I have a test tomorrow dealing with the factoring of trinomials, difference of squares, and sum/difference of cubes.

 

[mathwonk]

well a polynomila of degree three witha factor must have a factor of degree one.

but a polynomila has a factor x-a of degree one if and only if it has a root equal to a. now rational roots of integer polynomilas must be of form c/d where c is a factor of the constant term and d is a factor of the elading coefficient.


so that helps find all factors of integer polynomilas with rational coefficients.


so learn the "root / factor" theorem.l

 

[robert Ihnot]

The difference of squares is simple: a^2-b^2= (a-b)(a+b).

With cubes we have a^3-b^3=(a-b)(a^2+ab+b^2), and a^3+b^3= (a+b)(a^2-ab+b^2).

For trinominals, we look at things like (x-1)(x-6)=X^2-7x+6. You can use the root/factor theorem here. Simply solve for the roots of X^2-7x+6 = 0.
If its rational, it must be an integer since the leading term is unity, so we could try plus or minus: 1,2,3,6. You need practice on these kind of problems.

 

[wScott]

Well I didn't have time to look over your suggestions. But in any case I got a 24/25 on the test :D. the best thing is my teacher said it was one of the harder tests of the year.

I did good on the test but could you guys tell me what the root / factor theorem is? It sounds interesting.

 

[hypermorphism]

Mathwonk described the theorem in his post. For polynomials of the second degree, the result is trivial, so it is generally called upon to reduce higher degree polynomials with rational roots. Plug in rational combinations of factors of the constant term over factors of the leading coefficient until you get a zero, then use that root in a binomial factor to reduce the polynomial by one degree. Reduction is effected with long division of the larger polynomial by the binomial factor, if any rational factor exists.

For example, you want to find the zeros of the equation 2x^3 + 3x^2 - 4x - 1 = 0. By the theorem, if this equation has rational zeros, they can only be of the forms 1/2 and 1/1=1. Trying each term out, we find x=1 is a zero, so (x-1) is a factor of the polynomial. Long division gives 2x^3 + 3x^2 - 4x - 1 = (x-1)(2x^2 + 5x + 1). The quadratic is then easily factored to give the remaining irrational zeros of the original equation.

 

[mathwonk]

if you got 24/25 on the test and do not know the root factor theorem, hmmmm...

maybe by another name? it is a basic sophomore high school theorem that says a is a root iff x-a is a factor, ring a bell?

 

[wScott]

Nope, sorry mathwonk but I don't know that.

 

[mathwonk]

easy direction; if x-a is a facxtor then f(x) = (x-a)q(x), so seting x = a gives zero, so a is a root.

other direction: divide f(x) by x-a and get f(x) = (x-a)q(x) + r(x) where r is the remiander, and hence must have degree less than x-a, so muist have degree zero, i.e. r is a constant. then set x =a and get that r = f(a).

so x-a divides f if and only if the remainder on division by x-a equals zero, but since that remainder is f(a), x-a divides f(x) if and only if f(a) = 0, i.e. if and only if a is a root.

 

[quantumdude]

To wScott: Can I ask you something? Consider the expression $x^2-2$. It can be factored as $(x-\sqrt{2})(x+\sqrt{2})$. Have you ever seen anything like that before? Or would your teacher say that such an expression is not factorable?

To everyone else: Given the level of the questions being asked I have a strong hunch that wScott want to know when an expression is factorable over the integers. It's an unfortunate fact that today's math teachers refer to those expressions as simply "factorable" and all others as "not factorable", but that's the way it goes. For instance it is standard practice where I work for algebra instructors (not me of course) to refer to $x^2-2$, or even $x^2+2$ as not factorable, even though they are.