document.write( "Question 271956: Use binomial expansion to expand the following (2x-3y)^4 \n" ); document.write( "
Algebra.Com's Answer #199041 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "the other tutor's answer is incorrect\r
\n" ); document.write( "\n" ); document.write( "Use binomial expansion to expand the following (2x-3y)4
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document.write( "The first term is (2x) the second term is (-3y).\r\n" );
document.write( "The power is 4 so write \"(2x) (-3y)\" one more than 4, that is, 5 times:\r\n" );
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document.write( " (2x) (-3y)  +  (2x) (-3y)  +  (2x) (-3y)  +  (2x) (-3y)  +  (2x) (-3y) \r\n" );
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document.write( "Give the first factor (2x) a 4 power and the next (2x) a 3 power and so\r\n" );
document.write( "on down to the 0 power on the last term:\r\n" );
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document.write( " (2x)4(-3y)  +  (2x)3(-3y)  +  (2x)2(-3y)  +  (2x)1(-3y)  +  (2x)E(-3y) \r\n" );
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document.write( "Give the second factor (-3y) a 0 power and the next (-3y) a 1 power and so\r\n" );
document.write( "on up to the 4 power on the last term:\r\n" );
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document.write( " (2x)4(-3y)0 +  (2x)3(-3y)1 +  (2x)2(-3y)2 +  (2x)1(-3y)3 +  (2x)0(-3y)4\r\n" );
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document.write( "Give the first term the coefficient 1\r\n" );
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document.write( "1(2x)4(-3y)0 +  (2x)3(-3y)1 +  (2x)2(-3y)2 +  (2x)1(-3y)3 +  (2x)0(-3y)4\r\n" );
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document.write( "Multiply that coefficient 1 by the exponent of the first factor in the 1st\r\n" );
document.write( "term, which is 4, getting 4. Then divide that by the number of term which those\r\n" );
document.write( "came from, which is term number 1, so you divide 4 by 1 getting 4 and then write\r\n" );
document.write( "that 4 as the coefficient of the second term:\r\n" );
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document.write( "1(2x)4(-3y)0 + 4(2x)3(-3y)1 +  (2x)2(-3y)2 +  (2x)1(-3y)3 +  (2x)0(-3y)4\r\n" );
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document.write( "Multiply that coefficient 4 by the exponent of the first factor in the 2nd \r\n" );
document.write( "term, which is 3, getting 12. Then divide that by the number of term which those\r\n" );
document.write( "came from, which is term number 2, so you divide 12 by 2 getting 6 and then\r\n" );
document.write( "write that 6 as the coefficient of the third term:\r\n" );
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document.write( "1(2x)4(-3y)0 + 4(2x)3(-3y)1 + 6(2x)2(-3y)2 +  (2x)1(-3y)3 +  (2x)0(-3y)4\r\n" );
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document.write( "Multiply that coefficient 6 by the exponent of the first factor in the 3rd \r\n" );
document.write( "term, which is 2, getting 12. Then divide that by the number of term which those\r\n" );
document.write( "came from, which is term number 3, so you divide 12 by 3 getting 4 and then\r\n" );
document.write( "write that 4 as the coefficient of the fourth term:\r\n" );
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document.write( "1(2x)4(-3y)0 + 4(2x)3(-3y)1 + 6(2x)2(-3y)2 + 4(2x)1(-3y)3 +  (2x)0(-3y)4\r\n" );
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document.write( "Multiply that coefficient 4 by the exponent of the first factor in the 4th \r\n" );
document.write( "term, which is 1, getting 4. Then divide that by the number of term which those\r\n" );
document.write( "came from, which is term number 4, so you divide 4 by 4 getting 1 and then\r\n" );
document.write( "write that 4 as the coefficient of the fifth, and last term:\r\n" );
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document.write( "1(2x)4(-3y)0 + 4(2x)3(-3y)1 + 6(2x)2(-3y)2 + 4(2x)1(-3y)3 + 1(2x)0(-3y)4\r\n" );
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document.write( "Now we simplify. We erase the 1 coefficients and 1 exponents \r\n" );
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document.write( " (2x)4(-3y)0 + 4(2x)3(-3y)  + 6(2x)2(-3y)2 + 4(2x)(-3y)3  +  (2x)0(-3y)4\r\n" );
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document.write( "Anything to the zero power is 1 so we erase the factors with 0 exponents\r\n" );
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document.write( " (2x)4       + 4(2x)3(-3y)  + 6(2x)2(-3y)2 + 4(2x)(-3y)3  +       (-3y)4\r\n" );
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document.write( "We observe that negtive numbers raised to even powers are positive and\r\n" );
document.write( "negative numbers raised to odd powers are negative and so rewrite the\r\n" );
document.write( "signs as alternating: \r\n" );
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document.write( " (2x)4       - 4(2x)3(3y)   + 6(2x)2(3y)2  - 4(2x)(3y)3   +       (3y)4\r\n" );
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document.write( "  24x4       - 4*223x3*3y   +  6*22x2*32y2 -  4*2x*33y3   +        34y4\r\n" );
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document.write( "  16x4       - 4*8x3y       +  6*4x2*9y2   -   8x*27y3    +        81y4\r\n" );
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document.write( "  16x4       -  96x3y       +   216x2y2    -   216xy3     +        81y4\r\n" );
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document.write( "  16x4       -  96x3y       +   216x2y2    -   216xy3     +        81y4\r\n" );
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document.write( "Edwin

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