Worksheet For Quadratics With A Coefficient Greater Than 1 (Page 4 of 4) in pdf

Worksheet For Quadratics With A Coefficient Greater Than 1 Page 4

4. A student in factoring the original problem 2x

2 writes

(2x

4) x +

Is the student right or wrong? Explain.

This is an interesting question, as it depends on what you

mean by right or wrong. Multiplying out, we see that this is indeed

a factorization of the original. On the other hand, it not a factor-

ization into integer terms. The reason that we try to have standard

forms in mathematics is that we want to have a form that is easy to

recognize. Thus, when you are calculating limits or doing other sim-

pliﬁcation problems, it is easier to recognize terms that cancel if you

have a standard form. For polynomials, our standard form tends to be

that we factor out any integer factor of all terms, and then it is helpful

to make sure that the coeﬃcient of the highest power term is positive.

Thus ( 2)(x

3) is a standard form while (6

2x),

(2x

6) and

2(3

x) are not standard forms of ( 2x + 6). Thus the student can

be right, but neither term is in standard form if this is what is desired.

Moreover, if the problem was to fully factor, then they should have

factored out the 2 from (2x

4).

An important theorem from ring theory states that if a poly-

nomial with integer coeﬃcients factors over the rational numbers (that

is, has a factorization into non-constant polynomials with rational co-

eﬃcients), then it actually factors over the integers (so there are poly-

nomials with integer coeﬃcients). Thus, a correct factorization of a

polynomial with integer coeﬃcients into standard form, will never in-

volve fractions.

Worksheet For Quadratics With A Coefficient Greater Than 1 Page 4