FACTOR POLYNOMIALS by SPLITTING
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- Dorthy Gilbert
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1 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 + ax 2. 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 x 2 + 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 we SPLIT the MIDDLE. That is, the middle term is 5x but 5x can be written.. oohh.. so many ways... 5x = x+4x OR 5x = 20x 15x OR 7x 2x OR 2x+3x OR 100x+105x, etc... etc.. the possibilities are endless... but... Suppose we go on and split 5x as 5x = 2x + 3x then.. observe..., if we choose just the right way to split up 5x, we might just get lucky and proceed factor by grouping!!! x 2 + 5x + 6 x 2 + ( 5x ) + 6 = x 2 + ( 2x + 3x ) + 6 (BI, the creative, bold splitting the middle idea) = ( x 2 + 2x ) + ( 3x + 6 ) (ALA) = (x) [ x + 2 ] + (3) [ x + 2 ] (DL, BI, note: looks like someone distributed [x + 2]) = (x + 3) [ x + 2 ] Incidentally, rather than splitting 5x as 2x + 3x, we could have also split it up as 3x + 2x. Observe... x 2 + 5x + 6 x 2 + ( 5x ) + 6 = x 2 + ( 3x + 2x ) + 6 (BI, the creative, bold splitting the middle idea) = ( x 2 + 3x ) + ( 2x + 6 ) (ALA) = (x) [ x + 3 ] + (2) [ x + 3 ] (DL, BI, note: looks like someone distributed [x + 3]) = (x + 2) [ x + 3 ] Now, while this method has its limitation, it is a powerful one that can help factor more than degree two polynomials. Observe, 3x x c
2 3x 4 + ( 41x 2) + 60 = 3x 4 + ( 5x x 2) + 60 (BI, the creative, bold splitting the middle idea) = ( 3x 4 + 5x 2) + ( 36x ) (ALA) = (x 2 ) [ 3x ] + (12) [ 3x ] (DL, BI, note: looks like someone distributed [3x 2 + 5]) = (x ) [ 3x ] It even works with more than one variable, 6x yx y 2 6x 6 + ( 31yx 3) + 18y 2 = 6x 6 + ( 27yx 3 + 4yx 3) + 18y 2 (BI, the creative, bold splitting the middle idea) = ( 6x yx 3) + ( 4yx y 2) (ALA) = (3x 3 ) [ 2x 3 + 9y ] + (2y) [ 2x 3 + 9y ] (DL, BI, note: looks like someone distributed [2x 3 + 9y]) = (3x 3 + 2y) [ 2x 3 + 9y ] 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 5x for example, how would one know that 2x + 3x and 3x + 2x are good ways to split the middle and not 15x 10x nor 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 x 2 + 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 x 2 + 7x + 10 = (x + b)(x + d) For some appropriate constants a and b. Now we play detective to see how much we can guess about these unknown numbers a and b. If we work our way back, we can write (x + a)(x + b) = x 2 + ax + bx + ab (FOIL) = x 2 + (a + b)x + ab (ALA,DL) in theory this should be equal to our polynomial = x 2 + 7x + 10 c
![the creative, bold splitting the middle idea) = ( 6x 6 + 27yx 3) + ( 4yx 3 + 18y 2) (ALA) = (3x 3 ) [ 2x 3 + 9y ] + (2y) [ 2x 3 + 9y ] (DL, BI, note: looks like someone distributed [2x 3 + 9y]) = (3x](/docs-images/42/9183796/images/page_2.jpg "3 + 2y) [ 2x 3 + 9y ] 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.")
3 Here lies the secret on how to split up the 7 in 7x successfully. It should be split up into pieces a + b with the BIG hint that when you multiply ab you get the coefficient 10. 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 factor 10 = ab are: (1)(10) (-1)(-10) (2)(5) (-2)(-5) But, we also know that a + b = 7. This leads the elimination of most of these candidates for ab, leaving only the factors that add up to 7, thus the ones that will effectively help factor the polynomial. (1)(10) (-1)(-10) (2)(5) (-2)(-5) x 2 + ( 7x ) + 10 = x 2 + ( 2x + 5x ) + 10 (BI, THE split-the-middle idea!) = ( x 2 + 2x ) + ( 5x + 10 ) (ALA) = (x) [ x + 2 ] + (5) [ x + 2 ] (DL, BI, note: looks like someone distributed [x + 2]) = (x + 5) [ x + 2 ] 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 6x 2 + ( 19x ) + 10 First we multiply the leading coefficient, 6 times the last coefficient, 10 to obtain, 60. We now consider all the possible ways to factor 60 into the product of two integers: The ways to factor 60 = ab are: (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 for ab, leaving only the factors that add up to 19, 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) 6x 2 + ( 19x ) + 10 = 6x 2 + ( 4x + 15x ) + 10 (BI, THE split-the-middle idea!) = ( 6x 2 + 4x ) + ( 15x + 10 ) (ALA) = (2x) [ 3x + 2 ] + (5) [ 3x + 2 ] (DL, BI, note: looks like someone distributed [3x + 2]) = (2x + 5) [ 3x + 2 ] c
4 Yet another example Factor 10x 2 + ( 31x ) + 15 First we multiply the leading coefficient, 10 times the last coefficient, 15 to obtain, 150. We now consider all the possible ways to factor 150 into the product of two integers: The ways to factor 150 = ab are: (1)(150) (-1)(-150) (2)(75) (-2)(-75) (3)(50) (-3)(-50) (5)(30) (-5)(-30) (6)(25) (-6)(-25) (10)(15) (-10)(-15) But, we also know that a + b = 31. This leads the elimination of most of these candidates for ab, leaving only the factors that add up to 31, thus the ones that will effectively help factor the polynomial. (1)(150) (-1)(-150) (2)(75) (-2)(-75) (3)(50) (-3)(-50) (5)(30) (-5)(-30) (6)(25) (-6)(-25) (10)(15) (-10)(-15) 10x 2 + ( 31x ) + 15 = 10x 2 + ( 6x + 25x ) + 15 (BI, THE split-the-middle idea!) = ( 10x 2 + 6x ) + ( 25x + 15 ) (ALA) = (2x) [ 5x + 3 ] + (5) [ 5x + 3 ] (DL, BI, note: looks like someone distributed [5x + 3]) = (2x + 5) [ 5x + 3 ] Another example; Factor 2x 2 + ( 13x ) + 15 First we multiply the leading coefficient, 2 times the last coefficient, 15 to obtain, 30. We now consider all the possible ways to factor 30 into the product of two integers: The ways to factor 30 = ab are: (1)(30) (-1)(-30) (2)(15) (-2)(-15) (3)(10) (-3)(-10) (5)(6) (-5)(-6) (6)(5) (-6)(-5) But, we also know that a + b = 13. This leads the elimination of most of these candidates for ab, leaving only the factors that add up to 13, thus the ones that will effectively help factor the polynomial. (1)(30) (-1)(-30) (2)(15) (-2)(-15) (3)(10) (-3)(-10) (5)(6) (-5)(-6) (6)(5) (-6)(-5) c
5 2x 2 + ( 13x ) + 15 = 2x 2 + ( 3x + 10x ) + 15 (BI, THE split-the-middle idea!) = ( 2x 2 + 3x ) + ( 10x + 15 ) (ALA) = (x) [ 2x + 3 ] + (5) [ 2x + 3 ] (DL, BI, note: looks like someone distributed [2x + 3]) = (x + 5) [ 2x + 3 ] One more example; Factor 7x 2 + ( 31x ) + 12 First we multiply the leading coefficient, 7 times the last coefficient, 12 to obtain, 84. We now consider all the possible ways to factor 84 into the product of two integers: The ways to factor 84 = ab are: (1)(84) (-1)(-84) (2)(42) (-2)(-42) (3)(28) (-3)(-28) (4)(21) (-4)(-21) (6)(14) (-6)(-14) (7)(12) (-7)(-12) But, we also know that a + b = 31. This leads the elimination of most of these candidates for ab, leaving only the factors that add up to 31, thus the ones that will effectively help factor the polynomial. (1)(84) (-1)(-84) (2)(42) (-2)(-42) (3)(28) (-3)(-28) (4)(21) (-4)(-21) (6)(14) (-6)(-14) (7)(12) (-7)(-12) 7x 2 + ( 31x ) + 12 = 7x 2 + ( 3x + 28x ) + 12 (BI, THE split-the-middle idea!) = ( 7x 2 + 3x ) + ( 28x + 12 ) (ALA) = (x) [ 7x + 3 ] + (4) [ 7x + 3 ] (DL, BI, note: looks like someone distributed [7x + 3]) = (x + 4) [ 7x + 3 ] c
6 1. By splitting the middle term, factor the polynomial: x 2 + ( 2x ) By splitting the middle term, factor the polynomial: x 2 + ( 3x ) By splitting the middle term, factor the polynomial: x 2 + ( x ) By splitting the middle term, factor the polynomial: x 2 + (0) By splitting the middle term, factor the polynomial: x 2 + ( 4x ) By splitting the middle term, factor the polynomial: x 2 + ( x ) By splitting the middle term, factor the polynomial: 3x 2 + ( 7x ) By splitting the middle term, factor the polynomial: 6x 2 + ( 5x ) By splitting the middle term, factor the polynomial: x 2 + ( 2x ) By splitting the middle term, factor the polynomial: x 2 + ( 2x ) By splitting the middle term, factor the polynomial: 2x 2 + ( 3x ) By splitting the middle term, factor the polynomial: 2x 2 + ( 13x ) By splitting the middle term, factor the polynomial: 2x 2 + ( 13x ) + 15 c
7 14. By splitting the middle term, factor the polynomial: 6x 2 + ( x ) By splitting the middle term, factor the polynomial: 9x 2 + (0) By splitting the middle term, factor the polynomial: 6y 2 + ( 7y ) By splitting the middle term, factor the polynomial: 10y 2 + ( y ) By splitting the middle term, factor the polynomial: 15y 2 + ( 4y ) By splitting the middle term, factor the polynomial: 15x 4 + ( 13x 2) By splitting the middle term, factor the polynomial: 15x 4 + ( 8x 2) By splitting the middle term, factor the polynomial: 2x 4 + ( 7x 2) By splitting the middle term, factor the polynomial: 15y 2 + ( 16y ) By splitting the middle term, factor the polynomial: 3x 4 + ( x 2) By splitting the middle term, factor the polynomial: 3x 4 + ( 2x 2) By splitting the middle term, factor the polynomial: 2x 4 + ( 5x 2) By splitting the middle term, factor the polynomial: 2x 4 + ( 5yx 2) + 3y 2 c
8 27. By splitting the middle term, factor the polynomial: 2x 6 + ( 5x 3) By splitting the middle term, factor the polynomial: 2x 6 + ( y 4 x 3) + y By splitting the middle term, factor the polynomial: 9 + (0) + y By splitting the middle term, factor the polynomial: 25 + (0) + y By splitting the middle term, factor the polynomial: 9 + (0) + x By splitting the middle term, factor the polynomial: 5 + ( 4x 3) + x 6 c
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