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A special case: Factoring
"in pairs"
There
is one special case that you may or may not need, depending upon how
your book is structured and how your instructor intends to teach
factoring quadratics. I call it "factoring in pairs".
- Factor xy
– 5y – 2x + 10.
Is there anything that factors out of all four terms? No. When you have four terms, and nothing factors out of all of them,
think of "factoring in pairs". That is, split the expression into two pairs of terms, and factor
the pairs separately.
What can I factor out of the first pair? I can take out
a "y":
What can I factor out of the last pair? I can take out
a "–2":
xy – 5y – 2x + 10
= y(x
– 5) – 2x + 10
= y(x
– 5) – 2(x – 5)
(I took out a –2, rather than a 2, because the leading sign on the
pair was a "minus". And I got a "–5" in the result because, when I divided the positive 10 by the
negative 2, the result was a negative 5. Be careful with your signs!)
Now I do have a common factor, so I can proceed as usual:
xy – 5y – 2x
+ 10
= y(x – 5) – 2(x
– 5)
= (x – 5)(y
– 2)
Factoring "in pairs" is most commonly used to introduce
factoring quadratics. So you may see problems that look like this:
This is four terms with no common factor, so factor "in
pairs":
x2 + 4x – x
– 4
= x(x + 4) – 1(x + 4)
= (x
+ 4)(x – 1)
Note how, in the second line, I factored a "1" out. If "nothing"
factors out, a "1" factors out.
- Factor x2
– 4x + 6x – 24.
If you will be using "factoring in pairs" for factoring
quadratics, it will be called "factoring by grouping",
and it will work like this:
Find factors of –6 that add up to –5: use the number –6 and +1, because (–6)(+1)
= –6, and (–6)
+ (+1) = –5. Split the middle "–5x" term into "–6x"
and "+1x", and then factor in pairs:
x2 – 5x – 6
= x2 – 6x + 1x
– 6
= x(x – 6) + 1(x – 6)
= (x
– 6)(x + 1)
Find factors of (6)(6)
= 36 that add up to –13: use the numbers –9 and –4, because (–9)(–4) = 36, and (–9) + (–4) = –13. Split
the middle "–13x" term into "–9x" and "–4x", and factor in pairs:
6x2 – 13x +
6
= 6x2 – 9x
– 4x + 6
= 3x(2x – 3) – 2(2x
– 3)
= (2x
– 3)(3x – 2)
Related topics: factoring quadratics
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