question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

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If you multiply (x+2)(x+5) you get

x ( x + 5) + 2 ( x + 5) =

x^2 + 5 x + 2 x + 10.

You get x^2 from x * x, and this matches the x^2 in the given trinomial x^2 + 7 x + 10.

The 5x and the 2x add up to 7x, which matches the 7x in the given trinomial.

The 10, which came from 2 * 5, matches the 10 in the given trinomial.

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That verifies that the factoring is correct, but it doesn't completely address how you would find that result.

If a trinomial starting with just plain x^2 is to factor, then it has to factor into the form

(x + ?) * (x + ?),

because when you distribute this will give you the x^2.

Applying this to

x^2 + 7 x + 10

we need to find the ? values so that

(x + ?) ( x + ?) = x^2 + 7 x + 10.

The two ? mark values are the ones that will give you 10, when they are multiplied. The only ways to get 10 using integers are 1 + 10, -1 * (-10), 2 * 5 and -2 * (-5).

So if the trinomial factors it will be one of the following:

(x + 1) ( x + 10)

(x - 1) ( x - 10)

(x + 2) ( x + 5 )

(x - 2) ( x - 5).

It won't hurt right now to multiply all these out, but of course once we get used to the process we'll be able to see right away that some possibilities just won't work.

You should verify that

(x + 1) ( x + 10) = x^2 + 11 x + 10

(x - 1) ( x - 10) = x^2 - 11 x + 10

(x + 2) ( x + 5 ) = x^2 + 7 x + 10

(x - 2) ( x - 5) = x^2 - 7 x + 10.

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We see that the third possibility gives us the trinomial x^2 + 7 x + 10.

It is very possible that a given trinomial won't factor. For example if the given trinomial had been x^2 + 5 x + 10, none of the possibilities would have worked.

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There is more to the factoring of trinomials, but this should help you get through the most basic problems. We can address more difficult problems later, if you need to do so (and if you ask).

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