FACTOR POLYNOMIALS by SPLITTING

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FACTOR POLYNOMIALS by SPLITTING

THE IDEA FACTOR POLYNOMIALS by SPLITTING

The idea is to split the middle term into two pieces. Say the polynomial looks like

c+bx+ax2

. Further more suppose the DL METHOD fails, as well as the GROUPING METHOD... thus far these are the only two methods we have learned to factor polynomials, yet it is not enough. Consider for example

x2+ 5x+ 6

on this polynomial DL METHOD to factor fails since the gcd of the terms is 1, moreover, GROUPING METHOD fails, since all grouping combinations lead nowhere. In deed, it is time for a new idea. Suppose we go on a do something amazingly creative and bold. Suppose weSPLIT the MIDDLE. That is, the middle term is 5xbut 5x

can be written.. oohh.. so many ways.... 5x=x+ 4xOR 5x= 20x−15xOR 7x−2xOR 2x+ 3xOR−100x+ 105x, etc... etc.. the possibilities are endless.... but... Suppose we go on andsplit 5xas 5x= 2x+ 3xthen.. observe..., if we choose just the right way to split up 5x, we might just get lucky and proceed factor by grouping!!!

solution:

x2

+ 5x+ 6

x2+ 5x

+ 6 (given)

=x2

+ 2x+ 3x

+ 6 (BI, the creative, boldsplitting the middle idea) = x2+ 2x + 3x+ 6

(ALA) = (x)

x+ 2

+ (3) x+ 2

(DL, BI, note: looks like someone distributed[x+ 2]) = (x+ 3)

x+ 2

(DL)

Incidentally, rather than splitting5xas 2x+ 3x, we could have also split it up as3x+ 2x. Observe...

solution:

x2+ 5x+ 6

x2 + 5x

+ 6 (given)

=x2+ 3x+ 2x

+ 6 (BI, the creative, boldsplitting the middle idea) = x2+ 3x

+ 2x+ 6

(ALA) = (x)

x+ 3

+ (2) x+ 3

x+ 3

(DL)

Now, while this method has its limitation, it is a powerful one that can help factor more than degree two polynomials. Observe,

solution:

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3x4+ 41x2

+ 60 (given)

= 3x4+ 5x2

+ 36x2

+ 60 (BI, the creative, boldsplitting the middle idea) = 3x4+ 5x2 + 36x2+ 60

(ALA) = (x2)

3x2 + 5

+ (12) 3x2

+ 5

(DL, BI, note: looks like someone distributed[3x2 + 5]) = (x2+ 12)

3x2 + 5

(DL)

It even works with more than one variable,

solution:

6x6+ 31yx3+ 18y2

6x6+ 31yx3

+ 18y2 (given)

= 6x6+ 27yx3

+ 4yx3

+ 18y2 (BI, the creative, boldsplitting the middle idea)

= 6x6+ 27yx3 + 4yx3+ 18y2 (ALA)

= (3x3) 2x3

+ 9y

+ (2y) 2x3

+ 9y

(DL, BI, note: looks like someone distributed[2x3 + 9y]) = (3x3+ 2y)

2x3 + 9y

(DL)

Of course, at this point, the author is enjoying a tiny charlatan moment carrying on the mysteriously effective splitting-the-middle strategy and ignoring the 1000 lbs. gorilla in the discussion. Namely, amongst the millions of ways to split up 5xfor example, how would one know that 2x+ 3xand 3x+ 2xare good ways to split the middle and not 15x−10xnor the other millions of ways to split 5x??

Some Hints

1. This works well for trinomials (three terms)

2. Write the polynomial in descending order (of the exponet on the undeterminant) 3. If there is a common factor pull it out first (using D.L.)

THE IDEA behind THE IDEA: HOW to SPLIT the MIDDLE Consider the polynomial

x2

+ 7x+ 10

the secret to the splitting the middle method is to try to see a few moves ahead. Imagine we have already split up the middle, and grouped and factored so that

x2+ 7x+ 10 = (x+b)(x+d)

For some appropriate constantsaandb. Now we play detective to see how much we can guess about these unknown numbersaandb. If we work our way back, we can write

(x+a)(x+b) =x2+ax+bx+ab (FOIL)

=x2+ (a+b)x+ab (ALA,DL)

in theory this should be equal to our polynomial =x2+7x+10

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Here lies the secret on how to split up the 7 in 7xsuccessfully. It should be split up into pieces a+b with the BIG hint that when you multiplyabyou get the coefficient10. Now there aren’t that many ways to multiply two integers to get 10. We can list them all,

First we multiply the leading coefficient, 1 times the last coefficient, 10 to obtain, 10. We now consider all the possible ways to factor 10 into the product of two integers:

The ways to factor10 =abare: (1)(10) (-1)(-10) (2)(5) (-2)(-5)

But, we also know thata+b=7. This leads the elimination of most of these candidates for ab, leaving only the factors that add up to7, thus the ones that will effectively help factor the polynomial.

(1)(10) (-1)(-10) (2)(5) (-2)(-5)

Thus..

x2+ 7x

+ 10 (given)

=x2

+ 2x+ 5x

+ 10 (BI, THE split-the-middle idea!)

= x2+ 2x + 5x+ 10

(ALA) = (x)

x+ 2

+ (5) x+ 2

x+ 2

(DL)

The method just needs a slight modification when the leading coefficient is not 1. In this case, we multiply the leading coefficient and the last coefficient, and the factors of this product become the candidates to split the middle term. For example:

Factor

6x2+ 19x + 10

solution:

The ways to factor60 =abare:

(1)(60) (-1)(-60) (2)(30) (-2)(-30) (3)(20) (-3)(-20) (4)(15) (-4)(-15) (5)(12) (-5)(-12) (6)(10) (-6)(-10)

But, we also know that a+b=19. This leads the elimination of most of these candidates forab, leaving only the factors that add up to19, thus the ones that will effectively help factor the polynomial.

(1)(60) (-1)(-60) (2)(30) (-2)(-30) (3)(20) (-3)(-20)

(4)(15) (-4)(-15) (5)(12) (-5)(-12) (6)(10) (-6)(-10)

Thus..

6x2+ 19x

+ 10 (given)

= 6x2+ 4x+ 15x

= 6x2+ 4x

+ 15x+ 10

(ALA) = (2x)

3x+ 2

+ (5) 3x+ 2

(DL, BI, note: looks like someone distributed[3x+ 2]) = (2x+ 5)

3x+ 2

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Yet another example

Factor

10x2+ 31x + 15

solution:

(1)(150) (-1)(-150) (2)(75) (-2)(-75) (3)(50) (-3)(-50) (5)(30) (-5)(-30) (6)(25) (-6)(-25)

(10)(15) (-10)(-15)

(1)(150) (-1)(-150) (2)(75) (-2)(-75) (3)(50) (-3)(-50) (5)(30) (-5)(-30) (6)(25) (-6)(-25)

(10)(15) (-10)(-15)

Thus..

10x2

+ 31x

+ 15 (given)

= 10x2+ 6x+ 25x

= 10x2+ 6x

+ 25x+ 15

(ALA) = (2x)

5x+ 3

+ (5) 5x+ 3

(DL, BI, note: looks like someone distributed[5x+ 3]) = (2x+ 5)

5x+ 3

(DL)

Another example; Factor

2x2+ 13x + 15

solution:

(1)(30) (-1)(-30) (2)(15) (-2)(-15) (3)(10) (-3)(-10) (5)(6) (-5)(-6) (6)(5) (-6)(-5)

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Thus..

2x2+ 13x

+ 15 (given)

= 2x2+ 3x+ 10x

= 2x2 + 3x

+ 10x+ 15

(ALA) = (x)

2x+ 3

+ (5) 2x+ 3

(DL, BI, note: looks like someone distributed[2x+ 3]) = (x+ 5)

2x+ 3

(DL)

One more example; Factor

7x2+ 31x + 12

solution:

(1)(84) (-1)(-84) (2)(42) (-2)(-42) (3)(28) (-3)(-28) (4)(21) (-4)(-21) (6)(14) (-6)(-14)

(7)(12) (-7)(-12)

(1)(84) (-1)(-84) (2)(42) (-2)(-42) (3)(28) (-3)(-28) (4)(21) (-4)(-21) (6)(14) (-6)(-14)

(7)(12) (-7)(-12)

Thus..

7x2+ 31x

+ 12 (given)

= 7x2

+ 3x+ 28x

= 7x2+ 3x + 28x+ 12

(ALA) = (x)

7x+ 3

+ (4) 7x+ 3

(DL, BI, note: looks like someone distributed[7x+ 3]) = (x+ 4)

7x+ 3

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1. Bysplitting the middle term, factor the polynomial:

x2+ 2x + 1 2. Bysplitting the middle term, factor the polynomial:

x2

+ −x + −2 4. Bysplitting the middle term, factor the polynomial:

x2+(0)+ −1

x2+ x+ −6

7. Bysplitting the middle term, factor the polynomial: 3x2

+ 7x + −6 8. Bysplitting the middle term, factor the polynomial:

6x2+ 5x + −6 9. Bysplitting the middle term, factor the polynomial:

x2+ 2x + −8 10. Bysplitting the middle term, factor the polynomial:

x2

+ −2x + −8 11. Bysplitting the middle term, factor the polynomial:

2x2+ 3x + 1 12. Bysplitting the middle term, factor the polynomial:

2x2+ 13x + 15 13. Bysplitting the middle term, factor the polynomial:

−2x2+ 13x

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−6x2+ −x + 15 15. Bysplitting the middle term, factor the polynomial:

9x2+(0)+ −25 16. Bysplitting the middle term, factor the polynomial:

6y2

+ −7y + −5 17. Bysplitting the middle term, factor the polynomial:

10y2+ y + −2 18. Bysplitting the middle term, factor the polynomial:

15y2+ 4y + −4 19. Bysplitting the middle term, factor the polynomial:

15x4+ 13x2 + 2 20. Bysplitting the middle term, factor the polynomial:

15x4

+ 8x2 + 1 21. Bysplitting the middle term, factor the polynomial:

2x4+ 7x2 + 3 22. Bysplitting the middle term, factor the polynomial:

−15y2+ 16y + −4 23. Bysplitting the middle term, factor the polynomial:

3x4 + x2

+ −2 24. Bysplitting the middle term, factor the polynomial:

−3x4+ 2x2 + 1 25. Bysplitting the middle term, factor the polynomial:

2x4+ 5x2 + −3 26. Bysplitting the middle term, factor the polynomial:

2x4+ 5yx2

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2x6+ 5x3 + −3 28. Bysplitting the middle term, factor the polynomial:

2x6+ y4x3+ −y8

9 +(0)+ −y8 30. Bysplitting the middle term, factor the polynomial:

25 +(0)+ −y6

9 +(0)+ −x8

32. Bysplitting the middle term, factor the polynomial: 5 + 4x3