Download Factoring: Trinomials with Positive Coefficients

Factoring: Trinomials with Positive Coefficients
Main Idea To factor trinomials (polynomials with three terms), reverse
FOIL.
Remark 1 The concept behind factoring trinomials is simple; however, in practice
it gets tricky. Let’s make sure we understand what“reverse FOILing”
means.
→ FOIL→
ex.
−→
(2x + 3)..(x + 4) = 2x2 + .11x + 12 ..
← Reverse FOIL ←
So, to factor a trinomial like 2x2 + 11x + 12 we should think, “what
two binomials would I have to FOIL to get 2x2 + 11x + 12.” In other
words, we would have to fill in the blanks:
2x2 + 11x + 12 = (
+
)(
+
)
Remark 2 If we are going to reverse FOIL successfully we have to completely
understand regular FOILing. So let’s dissect it again.
first terms last terms
ex.
= 2x2 +8x+3x+12
−→ (2x + 3) · (x + 4) = (2..x + 3..) · (..x + .4) ..
↘↙
inner terms
2
= 2x +11x+12
. outer terms
What does this mean about factoring? To factor the trinomial
2x2 + 11x + 12, we would have to fill in the blanks using the following
reasoning:
Multiply to 2x2 Multiply to 12
2x2 + 11x + 12 = ( .
.
+ .
)( .
product of
inner terms
product of
+ outer terms
11x
+ .
) ..
Remark 3 Using this method, factoring trinomials becomes a guessing game.
We try to fill in the blanks following the rules of FOILing. To
help keep the rules and our guesses straight, we use a box like the
one below. The dashed arrows keep track of the rules of FOILing.
Procedure Factoring trinomials (with positive coefficients) of the form
ax2 + bx + c using “boxes”:
1 List all the factors of a.
2 List all the factors of c.
3 Draw a 2-by-2 box (see above). Place a pair of factors of a (step
1) in the left column of the box and place a pair of factors of
c (step 2) in the right column of the same box.
4 Multiply the diagonals in the box. If these diagonals can be
added to get b (the number in front of x in the original
polynomial), then the numbers in the top row and bottom row
give the factored form of the polynomial.
Remark 4 This procedure will not make sense until you see it and try it yourself!
Here is an example.
ex.
−→ Factor 2x2 + 15x + 18
steps 1 & 2
−→
steps 3 & 4
↓
The final box checks out, so we can read our factored form from it. The top row gives
the coefficients for the first binomial factor, and the bottom row gives the coefficients
for the second: 2x2 + 15x + 18 = (2x + 3)(1x + 6) = (2x + 3)(x + 6)
Example 1 Factor each trinomial.
a 3x2 + 10x + 7
b 5z 2 + 12z + 4
c 24y 2 + 62y + 33
d 6 + 7x + x2