Looking for PatternsDate: 10/30/2001 at 17:47:00 From: Amanda Subject: (no subject) What would be the answer to: (x-a)(x-b)(x-c)(x-d)(x-e)...(x-y)(x-z)? Thanks. Date: 10/31/2001 at 02:07:23 From: Doctor Jeremiah Subject: Re: (no subject) Hi Amanda, It depends. Do we know the values of any of these letters? Are any of them multiples of any of the others? If they are all different, and are not multiples of each other, then we can look for a pattern by starting small and working our way up: (x-a)(x-b) = x^2 - ax - bx + ab (x-a)(x-b)(x-c) = x^3 - cx^2 - ax^2 - bx^2 + cax + bcx + abx - abc (x-a)(x-b)(x-c)(x-d) = x^4 - dx^3 - cx^3 - ax^3 + dcx^2 + adx^2 - bx^3 + dbx^2 + cax^2 + bcx^2 + abx^2 - dcax - dbcx - dabx - abcx + dabc See the pattern? For 22 terms the first line would be x^22, and then every combination of 1 letter (26 of them) times x^21 would be subtracted, and then every combination of 2 letters times x^20 would be added, and then every combination of 3 letters times x^19 would be subtracted, and so on and so on until in the last line every combination of 26 letters (1 of them) times x^0 (which is 1) would be added. Try writing it down. It will be enormous! That's why it's always easier to look for patterns and extrapolate. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ Date: 11/14/2001 at 20:47:23 From: Natalie Subject: Value of an expression What is the value of the following expression: (x-a)(x-b)(x-c)...(x-z) so that there are a total of 26 factors, with each letter of the alphabet subtracted from x in one of the factors? I have tried to answer it and so have my parents and friends, but we are all confused. Please help! Date: 11/15/2001 at 16:19:56 From: Doctor Greenie Subject: Re: Value of an expression Hi, Natalie - I hope this is not a trick question. It is possible that the answer is 0, because one of the factors is (x-x), which is 0, and 0 multiplied by a bunch of other factors is still 0. I will assume it is not supposed to be a trick question, because it is kind of fun to try to imagine what the product of 26 binomials would look like. I don't know how you are going to write out the value of this expression. If you wrote it all out explicitly (showing every term), it would take probably a few dozen pages. Let's look at similar cases with smaller numbers of factors and see what those expressions look like in expanded form. Let's look first at the case with two factors.... (x-a)(x-b) = x^2 - ax - bx + ab This result comes from your familiar FOIL method of multiplying binomials. The FOIL method of multiplying binomials is a simplification of a general algorithm for multiplying polynomials. The general algorithm says that the product of any number of polynomials is the sum of all the possible "partial products" obtained by selecting one of the terms from each polynomial. For example, with the FOIL method for multiplying two binomials, we have F ("first") = first term * first term O ("outer") = first term * second term I ("inner") = second term * first term L ("last") = second term * second term Where does the x^2 term come from in this product? It comes from selecting the x term in each of the two factors. Where do the two x terms come from? They come from selecting the x term from one of the two factors and the constant term (-a or -b) from the other factor. And where does the constant term come from? It comes from selecting the constant terms from both factors. So let's rewrite this product as (x-a)(x-b) = x^2 - (a+b)x + ab In this expression, we have... (1) an x^2 term with coefficient 1; (2) an x term with a coefficient which is the sum of the constants in the two factors; and (3) a constant term equal to the product of the constants in the two factors Now let's look at the case with three factors. (x-a)(x-b)(x-c) = ? Where will the x^3 term(s) come from in the product? From selecting the x term from each of the three factors. This gives us a term x^3 in the product. Where will the x^2 terms come from in the product? From selecting the x term from two of the three factors and the constant from the other term. These terms give us -(a+b+c)x^2 in the product. Where will the x terms come from in the product? From selecting the x term from one of the three factors and the constant term from the other two. All the constant terms are negative; if I select two of them, then their product is positive. And selecting the constant factors in two of the three terms gives me the possible combinations ab, ac, and bc. So these terms give me +(ab+ac+bc)x in the product. And the constant term comes from selecting the constant term in each of the three factors. This gives me a term -abc in the product. So this product will be (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc And let's look at one more case, with four factors.... (x-a)(x-b)(x-c)(x-d) = ? Where will the x^4 term(s) come from in the product? From selecting the x term from each of the four factors. This gives us a term x^4 in the product. Where will the x^3 terms come from in the product? From selecting the x term from three of the four factors and the constant from the other term. These terms give us -(a+b+c+d)x^3 in the product. Where will the x^2 terms come from in the product? From selecting the x term from two of the four factors and the constant term from the other two. All the constant terms are negative; if I select two of them, then their product is positive. And selecting the constant factors in two of the four terms gives me the possible combinations ab, ac, ad, bc, bd, and cd. So these terms give me +(ab+ac+ad+bc+bd+cd)x^2 in the product. Where will the x terms come from in the product? From selecting the x term from 1 of the 4 factors and the constant term from the other three. All the constant terms are negative; if I select three of them, then their product is negative. And selecting the constant factors in three of the four terms gives me the possible combinations abc, abd, acd, and bcd. So these terms give me -(abc+abd+acd+bcd)x in the product. And the constant term comes from selecting the constant term in each of the 4 factors. This gives me a term abcd in the product. So this product will be (x-a)(x-b)(x-c)(x-d) = x^4 - (a+b+c+d)x^3 + (ab+ac+ad+bc+bd+cd)x^2 - (abc+abd+acd+bcd)x + abcd Now let's generalize the pattern. In the product of n factors (x-a)(x-b)(x-c)(x-d)(x-e)... we will get (1) terms of x^n, x^(n-1), x^(n-2), ..., x^2, x, and a constant term; (2) the terms in decreasing powers of x will have the x^n term positive, with the coefficients of successive terms having alternating signs; (3) the x^n term will come from selecting the x term from all n factors; there is only one way to do this, so the coefficient of the x^n term is 1; (4) the x^(n-1) term will come from selecting the x term from n-1 of the factors and the constant term from the other, so the coefficient of the x^(n-1) term will be -(sum of all the "products" of the constant terms taken 1 at a time); (5) the x^(n-2) term will come from selecting the x term from n-2 of the factors and the consant term from the other 2, so the coefficient of the x^(n-2) term will be +(sum of all the "products" of the constant terms taken 2 at a time); ... (?) the x^2 term will come from selecting the x term from two of the factors and the consant term from the other (n-2) terms, so the coefficient of the x^2 term will be (+ or -)(sum of all the "products" of the constant terms taken (n-2) at a time); (?) the x term will come from selecting the x term from 1 of the factors and the consant term from the other (n-1) terms, so the coefficient of the x term will be (+ or -)(sum of all the "products" of the constant terms taken (n-1) at a time); (?) the constant term will come from selecting the constant term in each factor, so the coefficient will be (+ or -)(product of all of the n constant terms) So the "answer" to your question is the following: (x-a)(x-b)(x-c)...(x-y)(x-z) = x^26 - (sum of all "products" of constant terms taken 1 at a time)x^25 + (sum of all products of constant terms taken 2 at a time)x^24 - (sum of all products of constant terms taken 3 at a time)x^23 + (sum of all products of constant terms taken 4 at a time)x^22 ... + or - ... ... + (sum of all products of constant terms taken 24 at a time)x^2 - (sum of all products of constant terms taken 25 at a time)x + (product of constant terms taken 26 at a time) The total number of terms in the expansion is 2^26 = 67108864 (because there are two choices for the term to select in each of the 26 factors), so it is clearly unreasonable to write out the expansion explicitly. I hope this helps. Write back if you have any further questions on this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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