13. Factor.

2x2 + 14x + 12

In this case, there is a common factor that we can factor out.

2(x2 + 7x + 6)

The expression inside the parentheses is then one of the easier factoring problems.

2(x + 6)(x + 1)

If we do not factor out the common factor first, the problem can still be done. Multiply the first and last coefficients   2x12 = 24,   and find a pair that add up to the middle coefficient.

The FOIL step, then, looks like

2x2 + 12x + 2x + 12

This will factor as

(2x + 2)(x + 6)

which looks a little different from out other answer. But if you notice, both of the terms in the first binomial factor have a common factor of   2   which can be factored out,

2(x + 1)(x + 6)

leaving us with the same factors we got when we did the problem the other way.

If there is a common factor that can be factored out, one will have a choice between whether to factor it out at the beginning or to factor it out later. You will find that it will make your job easier if you factor out the common factor at the beginning of the problem.

An interesting fact about polynomials is that if you factor a polynomial and keep factoring the factors until you get fctors that can not be factored any farther, then, while the factors may not be in the same order, multiplication is commutative, there will be the same numbers of the same factors.

This is called unique factorization of polynomials.

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