In this case the coefficient on the first term is not just an
invisible 1. So here when we multiply the first and last coefficients,
we get something different than the last term. (2)(15) = 30, so we
are looking for factors of 30 that add up to 13. Many students will be
able to immediately recognize that 3 and 10 multiply up to 30 and
add up to 13, but if the answer does not immediately pop into your
head, there is something you can be doing with your hands while we
wait for it to happen. Write down all of the factorizations of 30 into
a product of two numbers starting with 1 time 30.
When we divide 1, 2, and 3 into 30, we find that they go into it,
and we get the first three results in our table. After 1, 2, and 3, we
get to 4. 4 doesn't go into 30, so there is no entry for 4. After 4 comes
5 which does go into 30, and it appears in the table. After 5 comes 6,
but we see a 6 in the left column of our table. As soon as you find
yourself considering a number which appears in the left side of the
table, you know that you have gotten all of the possibilities, because
if we were to find a number bigger than 6 which went into 30, its
quotient would be smaller than 5, and we were very thorough in
already considering all of those possibilities. Fortunately, we see that
there is a pair of numbers in the table that add up to the middle
coefficient, namely 10 and 3. That gives us the coefficients of the O
and I terms in the FOIL step.
Factor out the greatest common factor in the F and O terms.
That will be the first term in the first binomial.
We can now figure that the other First term must be an x.
and the other Outer term must b a -5.
There are two ways of looking at the final term. It is one of the
Last terms, and it is one of the Inner terms. There is something
which will satisfy both requirements, namely -3.
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