SOLUTION: Factor the expression completely. (This type of expression arises in calculus in using the "product rule.") {{{3x^2 (4x-12)^2 + x^3 * 2(4x-12)*4}}} The final answer is: {{

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor the expression completely. (This type of expression arises in calculus in using the "product rule.") {{{3x^2 (4x-12)^2 + x^3 * 2(4x-12)*4}}} The final answer is: {{      Log On


   

Question 541285: Factor the expression completely. (This type of expression arises in calculus in using the "product rule.")
3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%284x-12%29%2A4
The final answer is:
16x%5E2%28x-3%29%285x-9%29
Please show detailed step to obtain the answer so i can follow what you are doing.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given to factor:
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3x%5E2+%284x-12%29%5E2+%2B+x%5E3+%2A+2%2A%284x-12%29%2A4
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I'm not sure that this is the most efficient way to get to the answer, but here's what I did:
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I started by breaking down the exponent terms into their factors:
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3%2Ax%2Ax%2A%284x-12%29%2A%284x-12%29+%2B+x%2Ax%2Ax%2A2%2A%284x+-12%29+%2A4
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Next I noted that the term (4x - 12) can be factored to 4*(x - 3). So I replaced each (4x - 12) with 4*(x - 3) to get:
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3%2Ax%2Ax%2A4%2A%28x+-+3%29%2A4%2A%28x+-3%29+%2B+x%2Ax%2Ax%2A2%2A4%2A%28x+-+3%29%2A4
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Then I pulled x*x from each of the two terms (the two terms are separated by the + sign):
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x%2Ax%283%2A4%2A%28x-3%29%2A4%2A%28x-3%29%2B%28x%2A2%2A4%2A%28x-3%29%2A4%29%29
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Next I pulled out an (x-3) factor from both the terms:
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x%2Ax%2A%28x+-3%29%283%2A4%2A4%2A%28x-3%29%2Bx%2A2%2A4%2A4%29
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Then I multiplied the constants. First to find that 3*4*4 = 48. Then to find that 2*4*4 = 32. This made the resulting expression become:
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x%2Ax%2A%28x-3%29%2A%2848%2A%28x-3%29%2B+32%2Ax%29
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Then I factored 16 out of each of the two terms in parentheses to get:
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x%2Ax%2A%28x-3%29%2A16%283%2A%28x-3%29%2B+2x%29
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Next I multiplied out the 3 times (x-3) to get 3x - 9 inside the parentheses and get:
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x%2Ax%2A%28x-3%29%2A16%2A%283x+-+9+%2B+2x%29
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Inside the parentheses I combined the 3x and 2x into 5x so that the expression became:
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x%2Ax%2A%28x-3%29%2A16%2A%285x+-9%29
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I then substituted x^2 for x*x and moved the 16 to the front of the line to get:
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16%2Ax%5E2%2A%28x-3%29%2A%285x+-9%29
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and that's the answer that you were looking for. Hope this helps you to see a way to do this problem.
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Regarding your question: When we factor (4x-12)^2 it is just (4x-12)(4x-12) instead of (4x-12)(4x+12)? According to the factoring formula "Difference of squares", (A-B)^2 = (A-B)(A+B). Guess this is where i got lost but don't understand when you don't foil out these binomials.
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You're a little confused about factoring the difference of squares. The factoring formula says:
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A^2 - B^2 = (A - B)*(A + B)
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Notice the left side of this rule says A^2 - B^2. (A - B)^2 which for this problem is (4x - 12)^2 is not the same as A^2 - B^2. If you have (4x - 12)^2 and you square it by FOIL multiplying (4x - 12)*(4x - 12) you get that:
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(4x - 12)^2 = 16x^2 - 96x + 144.
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In general, when you have (A - B)^2 and you square it the result is A^2 - 2AB + B^2 and you can see that it is different from A^2 - B^2 because it has a middle term -2AB and also because it has a positive B^2 instead of a minus B^2. Hope this helps you to see why the difference of squares rule for factoring does not apply to (4x - 12)^2. It would apply if we had (4x)^2 - (12)^2, but that was not the case in this problem.