Algebra
Tyler C.
asked 08/20/12Please show me work of how to solve this problem. I've been beating myself up! Thanks.
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Kristin R. answered 08/20/12
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(x+y)² =(x+y)(x+y) Then you FOIL (First, outer, inner, last)
(x+y)² =(x+y)(x+y) =xx + xy + xy + yy [and when you combine like terms] =x² + 2xy + y²
(x+y)3 = (x² + 2xy + y²)(x+y) Then you FOIL (First, outer, inner, last)
(x+y)3 = (x² + 2xy + y²)(x+y) = x2x +2xxy + xy2 + x2y + 2xyy + y2y[and when you combine like terms]=x3 + 3x2y+ 3xy2 + y3
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Tyler C.
Thank you, Kristin! How did you make exponents on the computer? I couldn't figure that out.
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08/20/12
Adam B.
Use Microsoft Word and hold the control, shift, and equals key for superscript. Hold the control and equal key only for subscript.
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08/23/12
Cathy T.
If that's the case, shouldn't you get the original expression- (x+3)^3- as your answer after you factored your answer -x3 + 3x2y+ 3xy2 + y3?
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06/26/18
Lily W.
"Tyler C." Here's a good tip to make exponents click ⌘+control+space. It will pop up emojis. On the upper right-hand corner, u'll see something that has the ⌘ icon. Click that. It will pull up the character viewer. Click the section that says "digits - all". {If u can't find "digits - all", you can click the settings icon on the left corner of the character viewer page and then click "custom list". You can find "digits - all" on that list} Anyway when u get to "digits - all", scroll down until you see tiny numbers. { ⁰ ¹ ² ³ ⁴ ₁ ₂ ₃ ₄} these are a few examples I picked out. These are subscripts and superscripts. When u click on the superscript u want, you can actually click the button that says "add to favorites". When u do this, you can click on the ⌘ icon again on the upper right-hand corner and go back to the emojis. Your "favorites" will be listed. This way, you can easily get to your "exponents" without having to go through all that trouble. Hope this helps!!Report
02/15/19
Paul T. answered 08/21/12
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To make the exponents, use the shortcut for superscripts: Press CTRL+SHIFT+=; then release the 3 keys and type whatever exponent you wish to insert.
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Raymond B. answered 08/15/19
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(x+y)3 expanded has 4 terms, 1 more than the exponent, x3 x2y xy2 and y3 x is decreasing from 3 to 0 from left to right, as y increases from 0 to 3. Any number or variable to the 0 power is 1.
Then you need the coefficients for each of the 4 terms. You can read that off Pascal's triangle:
1
11
121
1331
1, 3, 3 and 1 are the coefficients that go with each of the 4 terms to get x^3 +3x^2y + 3xy^2 + y^3
Or if you know combinations from probability theory, the coefficients are (3/3), (3/2), (3/1) and (3/0)
(n/r) = n!/r!(n-r!) (3/3) = 3!/3!0! = 1, (3/2)= 3!/2!1! = 3, (3/1) = 3!/1!2! = 3 and (3/0) = 3!/0!3! = 1
0! = 1, 1! = 1, 2! = 2x1 = 2, 3! =3x2x1 = 6
x3 + 3x2y + 3xy2 + y3
Each number in Pascal's triangle is the sum of the 2 digits just above it, to the right & left.
Or you could just multiply the 3 factors together (x+y)(x+y)(x+y) =(x^2+2y+y^2)(x+y)= (x^3+3x^2y+3xy^2+y^3)
Or you might just remember the formula: (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 and replace a with x and b with y
People remember the formula (a+b)^2 = a^2 +2ab + b^2, for the cubic it's similar
power of a decreases by 1 each term, 2, 1, then an implicit 0 while b increases by 1 from an implicit 0 to 1, to 2
for the cubic expression powers of a decrease from 3, to 2, to 1, to 0
while b increases from 0, to 1, to 2, to 3
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Adam B. answered 08/23/12
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If you want to do the caclulations alot quicker, use Pascal's triangle. Let's say you have (x+y)^n. If you take n! and divide it by (n-r)!*r! then you will have the coefficient to each part of the equation. ! is factorial. Lets say you have 5! = 5*4*3*2*1. r would equal the position of the coefficient. In the case of (x+y)^3 the numbers on pascals triangle are 1 3 3 1. Which means the answer is 1*x^3 + 3*(x^2)(y) + 3*(x)(y^2) + 1*y^3. Do you see a pattern with x and y. The numbers in pascals triangle represents how many different combinations can be taken from a limited number of something. (Ex. How many ways can you take 3 employees from 5 employees). The answer would be 10 ways if you use the equation. Here is a more detailed explanation of Factorial. Imagine you have five pieces of candy. If you choose one then you can only choose four more. If you choose another piece of candy then you can only choose three more pieces and so on and so on until you arrive at 1. (5*4*3*2*1). This method will save you so much time if you have to deal with large exponents. If you don't understand I will be happy to explain with more clarity.
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